book cover

KNOWLEDGE
GRAPHS

Aidan Hogan | Eva Blomqvist | Michael Cochez
Claudia d’Amato | Gerard de Melo | Claudio Gutierrez
Sabrina Kirrane | José Emilio Labra Gayo | Roberto Navigli
Sebastian Neumaier | Axel-Cyrille Ngonga Ngomo
Axel Polleres | Sabbir M. Rashid | Anisa Rula
Lukas Schmelzeisen | Juan Sequeda
Steffen Staab | Antoine Zimmermann

Knowledge Graphs

About the book

The book is published by Morgan & Claypool in the series Synthesis Lectures on the Data, Semantics, and Knowledge edited by Ying Ding and Paul Groth. Please, cite the book as:

Aidan Hogan, Eva Blomqvist, Michael Cochez, Claudia d’Amato, Gerard de Melo, Claudio Gutierrez, Sabrina Kirrane, José Emilio Labra Gayo, Roberto Navigli, Sebastian Neumaier, Axel-Cyrille Ngonga Ngomo, Axel Polleres, Sabbir M. Rashid, Anisa Rula, Lukas Schmelzeisen, Juan Sequeda, Steffen Staab, Antoine Zimmermann (2021) Knowledge Graphs, Synthesis Lectures on Data, Semantics, and Knowledge, No. 22, 1–237, DOI: 10.2200/S01125ED1V01Y202109DSK022, Morgan & Claypool

BibTeX entry of this book:

@book{kg-book,
  author = {Hogan, Aidan and Blomqvist, Eva and Cochez, Michael and
d'Amato, Claudia and de Melo, Gerard and Guti\'errez, Claudio and
Kirrane, Sabrina and Labra Gayo, Jos\'e Emilio and Navigli, Roberto and
Neumaier, Sebastian and Ngonga Ngomo, Axel-Cyrille and Polleres, Axel and
Rashid, Sabbir M. and Rula, Anisa and Schmelzeisen, Lukas and
Sequeda, Juan F. and Staab, Steffen and Zimmermann, Antoine},
  doi = {10.2200/S01125ED1V01Y202109DSK022},
  isbn = {9781636392363},
  language = {English},
  number = {22},
  numpages = {237},
  publisher = {Morgan \& Claypool},
  series = {Synthesis Lectures on Data, Semantics, and Knowledge},
  title = {{K}nowledge {G}raphs},
  url = {https://kgbook.org/},
  year = {2021}
}
ISBN paperback:
9781636392356
ISBN ebook:
9781636392363
ISBN hardcover:
9781636392370

Copyright © 2021 by Morgan & Claypool. All rights reserved.

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SYNTHESIS LECTURES ON ON DATA, SEMANTICS, AND KNOWLEDGE #22

Abstract

This book provides a comprehensive and accessible introduction to knowledge graphs, which have recently garnered notable attention from both industry and academia.

Knowledge graphs are founded on the principle of applying a graph-based abstraction to data, and are now broadly deployed in scenarios that require integrating and extracting value from multiple, diverse sources of data at large scale. The book defines knowledge graphs and provides a high-level overview of how they are used. It presents and contrasts popular graph models that are commonly used to represent data as graphs, and the languages by which they can be queried before describing how the resulting data graph can be enhanced with notions of schema, identity, and context. The book discusses how ontologies and rules can be used to encode knowledge as well as how inductive techniques — based on statistics, graph analytics, machine learning, etc. — can be used to encode and extract knowledge. It covers techniques for the creation, enrichment, assessment, and refinement of knowledge graphs and surveys recent open and enterprise knowledge graphs and the industries or applications within which they have been most widely adopted. The book closes by discussing the current limitations and future directions along which knowledge graphs are likely to evolve.

This book is aimed at students, researchers, and practitioners who wish to learn more about knowledge graphs and how they facilitate extracting value from diverse data at large scale. To make the book accessible for newcomers, running examples and graphical notation are used throughout. Formal definitions and extensive references are also provided for those who opt to delve more deeply into specific topics.

Keywords

knowledge graphs, graph databases, knowledge graph embeddings, graph neural networks, ontologies, knowledge graph refinement, knowledge graph quality, knowledge bases, artificial intelligence, semantic web, machine learning

Preface

The origins of this book can be traced back to a Dagstuhl Seminar, held in 2018, on the topic of Knowledge Graphs. At the time of the seminar, the topic was quickly becoming mainstream in academia and industry, but there were conflicting messages as to what a “knowledge graph” was. Much of the discussion of the seminar centred on this question, and there were divergent opinions as to how knowledge graphs could (or should) be defined; how they relate to previous concepts such as graph databases, knowledge bases, ontologies, RDF graphs, property graphs, semantic networks, etc.; and how the emerging area of Knowledge Graphs should be positioned with respect to the established areas of Artificial Intelligence, Big Data, Databases, Graph Theory, Logic, Machine Learning, Knowledge Representation, Natural Language Processing, Networks (in their various forms), and the Semantic Web. As the discussion continued, a consensus began to emerge: Knowledge Graphs, as a topic, involves a novel confluence of techniques stemming from previously disparate scientific communities, with the unifying goal of developing novel graph-based techniques for better integrating and extracting value from diverse knowledge sources at large scale.

As a follow-up to the seminar, the attendees agreed that in order to foster this unifying view of Knowledge Graphs, there was a need for a manuscript that would serve as a general introduction to the area. This manuscript would:

The manuscript would then serve as an introductory text for students, practitioners and researchers new to the area, helping to form a consensus in terms of what is a knowledge graph, laying the foundations for future developments.

The goal of preparing this manuscript was an ambitious one, and involved drawing together and distilling down a vast amount of literature on a diverse range of topics into a set of key concepts described in an accessible way. For this reason, the manuscript has been prepared by many authors, who have lent their knowledge and expertise to the preparation of specific sections. A short version of the manuscript was first published as a tutorial paper [Hogan et al., 2021], consisting of an abridged version of the first five chapters of this book, along with a summary of how knowledge graphs are used in practice, and conclusions. However, there was not enough space to describe all of the important developments in the area. This led us to publish this book, which further includes topics relating to the creation, enrichment, quality assessment, refinement and publication of knowledge graphs, as well as formal definitions, a historical perspective, and extended discussion throughout.

The book is divided into ten chapters. The first chapter provides a general introduction to the area, defines the concept of a “knowledge graph”, and provides a high-level overview of how knowledge graphs are currently being used. The second chapter presents and contrasts popular graph models that are commonly used to represent data as graphs, and the languages by which they can be queried. The third chapter describes how the resulting data graph can be enhanced with notions of schema, identity and context. The fourth chapter discusses how ontologies and rules can be used to encode knowledge, and how they enable deductive forms of reasoning. The fifth chapter delves into how inductive techniques – based on statistics, graph analytics, machine learning, etc. – can be used to encode and extract knowledge. The sixth chapter is dedicated to techniques for the creation and enrichment of knowledge graphs from legacy sources of data. The seventh chapter enumerates a variety of quality measures that can be used to assess a knowledge graph in terms of its fitness for use in a variety of applications. The eighth chapter presents key methods for the refinement of knowledge graphs, with the goal of improving their completeness and correctness. The ninth chapter provides a survey of the open and enterprise knowledge graphs that have emerged in recent years, along with the industries within which, and the applications for which, they have been most widely adopted. The tenth chapter wraps up the book with discussion of the current limitations and future directions along which knowledge graphs are likely to evolve. An appendix further covers knowledge graphs from an historical perspective, establishing their significance in the broader context of the academic study of data and knowledge, as well as surveying prior definitions of “knowledge graphs” from the literature.

A key aim of this book is to be accessible to a broader audience. While background knowledge of related topics such as Databases, Logic, Machine Learning, Semantic Web, etc., will help to understand some of the particular topics mentioned, such a background is not necessary to follow the general concepts described within. The book aims to motivate and illustrate the various concepts it introduces from a practical perspective, and in order to be as accessible as possible, relies heavily on an example-driven presentation using a graphical notation. For the reader wishing to dig more into the technical minutiae, we complement this discussion with formal definitions throughout; however, the reader more interested in understanding the general concepts and their rationale will find the discussion to be self-contained if they choose to skip the definitions presented in visually distinctive boxes.

The book serves as an entry point for those new to the topic, and may thus serve as a useful textbook for university courses, for researchers who are venturing into the topic for the first time, and for practitioners who wish to understand more about how knowledge graphs might be of use within their company or organisation, or indeed, how to maximise the value of the knowledge graphs that they are currently developing. Readers who are already active within specific sub-areas of Knowledge Graphs may further appreciate the technical definitions included, the references to other literature provided, and the broader perspective that this book offers in terms of the other related sub-areas and how they complement each other.

By drawing together diverse techniques from disparate areas, Knowledge Graphs has become an exciting topic in terms of both research and applications. We expect to see growing interest on this topic as the years advance, and indeed hope that this book will help to more firmly establish the foundations of this topic, and to foster future developments upon these foundations, potentially by its readers.

Aidan Hogan, Eva Blomqvist, Michael Cochez, Claudia d’Amato, Gerard de Melo, Claudio Gutierrez, Sabrina Kirrane, José Emilio Labra Gayo, Roberto Navigli, Sebastian Neumaier, Axel-Cyrille Ngonga Ngomo, Axel Polleres, Sabbir M. Rashid, Anisa Rula, Lukas Schmelzeisen, Juan Sequeda, Steffen Staab, Antoine Zimmermann
September 2021

Acknowledgements

We thank the organisers and attendees of the Dagstuhl Seminar on “Knowledge Graphs”. We also thank those who provided feedback on this content.

Hogan was funded by Fondecyt Grant No. 1181896. Hogan & Gutierrez were funded by ANID – Millennium Science Initiative Program – Code ICN17_002. Cochez did part of the work while employed at Fraunhofer FIT, Germany and was later partially funded by Elsevier’s Discovery Lab. Kirrane, Ngonga Ngomo, Polleres & Staab received funding through the project “KnowGraphs” from the European Union’s Horizon programme under the Marie Skłodowska-Curie grant agreement No. 860801. Kirrane & Polleres were supported by the European Union’s Horizon 2020 research and innovation programme under grant 731601. Labra was supported by the Spanish Ministry of Economy and Competitiveness (Society challenges: TIN2017-88877-R). Navigli was supported by the MOUSSE ERC Grant No. 726487 under the European Union’s Horizon 2020 research and innovation programme. Rashid was supported by IBM Research AI through the AI Horizons Network. Schmelzeisen was supported by the German Research Foundation (DFG) grant STA 572/18-1.

Aidan Hogan, Eva Blomqvist, Michael Cochez, Claudia d’Amato, Gerard de Melo, Claudio Gutierrez, Sabrina Kirrane, José Emilio Labra Gayo, Roberto Navigli, Sebastian Neumaier, Axel-Cyrille Ngonga Ngomo, Axel Polleres, Sabbir M. Rashid, Anisa Rula, Lukas Schmelzeisen, Juan Sequeda, Steffen Staab, Antoine Zimmermann
September 2021

Introduction

Though the phrase “knowledge graph” has been used in the literature since at least 1972 [Schneider, 1973], the modern incarnation of the phrase stems from the 2012 announcement of the Google Knowledge Graph [Singhal, 2012], followed by further announcements of knowledge graphs by Airbnb [Chang, 2018], Amazon [Krishnan, 2018], eBay [Pittman et al., 2017], Facebook [Noy et al., 2019], IBM [Devarajan, 2017], LinkedIn [He et al., 2016], Microsoft [Shrivastava, 2017], Uber [Hamad et al., 2018], and more besides. The growing industrial uptake of the concept proved difficult for academia to ignore: more and more scientific literature is being published on knowledge graphs, which includes books (e.g., [Pan et al., 2017, Qi et al., 2021, Fensel et al., 2020, Kejriwal et al., 2021]), as well as papers outlining definitions (e.g., [Ehrlinger and Wöß, 2016]), novel techniques (e.g., [Pujara et al., 2013, Wang et al., 2014, Lin et al., 2015]), and surveys of specific aspects of knowledge graphs (e.g., [Paulheim, 2017, Wang et al., 2017]).

Underlying all such developments is the core idea of using graphs to represent data, often enhanced with some way to explicitly represent knowledge [Noy et al., 2019]. The result is most often used in application scenarios that involve integrating, managing and extracting value from diverse sources of data at large scale [Noy et al., 2019]. Employing a graph-based abstraction of knowledge has numerous benefits in such settings when compared with, for example, a relational model or NoSQL alternatives. Graphs provide a concise and intuitive abstraction for a variety of domains, where edges capture the (potentially cyclical) relations between the entities inherent in social data, biological interactions, bibliographical citations and co-authorships, transport networks, and so forth [Angles and Gutierrez, 2008]. Graphs allow maintainers to postpone the definition of a schema, allowing the data – and its scope – to evolve in a more flexible manner than typically possible in a relational setting, particularly for capturing incomplete knowledge [Abiteboul, 1997]. Unlike (other) NoSQL models, specialised graph query languages support not only standard relational operators (joins, unions, projections, etc.), but also navigational operators for recursively finding entities connected through arbitrary-length paths [Angles et al., 2017]. Standard knowledge representation formalisms – such as ontologies [Hitzler et al., 2012, Brickley and Guha, 2014, Mungall et al., 2012] and rules [Horrocks et al., 2004, Kifer and Boley, 2013] – can be employed to define and reason about the semantics of the terms used to label and describe the nodes and edges in the graph. Scalable frameworks for graph analytics [Malewicz et al., 2010, Xin et al., 2013a, Stutz et al., 2016] can be leveraged for computing centrality, clustering, summarisation, etc., in order to gain insights about the domain being described. Various representations have also been developed that support applying machine learning techniques both directly and indirectly over graphs [Wang et al., 2017, Wu et al., 2019].

In summary, the decision to build and use a knowledge graph opens up a range of techniques that can be brought to bear for integrating and extracting value from diverse sources of data at large scale. The goal of this book is to motivate and give a comprehensive introduction to knowledge graphs: to describe their foundational data models and how they can be queried; to discuss representations relating to schema, identity, and context; to discuss deductive and inductive ways to make knowledge explicit; to present a variety of techniques that can be used for the creation and enrichment of graph-structured data; to describe how the quality of knowledge graphs can be discerned and how they can be refined; to discuss standards and best practices by which knowledge graphs can be published; and to provide an overview of existing knowledge graphs found in practice. Our intended audience includes researchers and practitioners who are new to knowledge graphs. As such, we do not assume that readers have specific expertise on knowledge graphs.

Knowledge graph. The definition of a “knowledge graph” remains contentious [Ehrlinger and Wöß, 2016, Bonatti et al., 2018, Bergman, 2019], where a number of (sometimes conflicting) definitions have emerged, varying from specific technical proposals to more inclusive general proposals; we address these prior definitions in Appendix A. Herein we adopt an inclusive definition, where we view a knowledge graph as a graph of data intended to accumulate and convey knowledge of the real world, whose nodes represent entities of interest and whose edges represent relations between these entities. The graph of data (aka data graph) conforms to a graph-based data model, which may be a directed edge-labelled graph, a property graph, etc. (we discuss concrete alternatives in Chapter 2). By knowledge, we refer to something that is known. Such knowledge may be accumulated from external sources, or extracted from the knowledge graph itself. Knowledge may be composed of simple statements, such as “Santiago is the capital of Chile”, or quantified statements, such as “all capitals are cities”. Simple statements can be accumulated as edges in the data graph. If the knowledge graph intends to accumulate quantified statements, a more expressive way to represent knowledge – such as ontologies or rules – is required. Deductive methods can then be used to entail and accumulate further knowledge (e.g., “Santiago is a city”). Additional knowledge – based on simple or quantified statements – can also be extracted from and accumulated by the knowledge graph using inductive methods.

Knowledge graphs are often assembled from numerous sources, and as a result, can be highly diverse in terms of structure and granularity. To address this diversity, representations of schema, identity, and context often play a key role, where a schema defines a high-level structure for the knowledge graph, identity denotes which nodes in the graph (or in external sources) refer to the same real-world entity, while context may indicate a specific setting in which some unit of knowledge is held true. As aforementioned, effective methods for extraction, enrichment, quality assessment, and refinement are required for a knowledge graph to grow and improve over time.

In practice. Knowledge graphs aim to serve as an ever-evolving shared substrate of knowledge within an organisation or community [Noy et al., 2019]. We distinguish two types of knowledge graphs in practice: open knowledge graphs and enterprise knowledge graphs. Open knowledge graphs are published online, making their content accessible for the public good. The most prominent examples – DBpedia [Lehmann et al., 2015], Freebase [Bollacker et al., 2007b], Wikidata [Vrandečić and Krötzsch, 2014], YAGO [Hoffart et al., 2011], etc. – cover many domains and are either extracted from Wikipedia [Lehmann et al., 2015, Hoffart et al., 2011], or built by communities of volunteers [Bollacker et al., 2007b, Vrandečić and Krötzsch, 2014]. Open knowledge graphs have also been published within specific domains, such as media [Raimond et al., 2014], government [Hendler et al., 2012, Shadbolt and O'Hara, 2013], geography [Stadler et al., 2012], tourism [Lu et al., 2016, Kärle et al., 2018, Maturana et al., 2018, Zhang et al., 2019], life sciences [Callahan et al., 2013], and more besides. Enterprise knowledge graphs are typically internal to a company and applied for commercial use-cases [Noy et al., 2019]. Prominent industries using enterprise knowledge graphs include Web search (e.g., Bing [Shrivastava, 2017], Google [Singhal, 2012]), commerce (e.g., Airbnb [Chang, 2018], Amazon [Krishnan, 2018, Dong, 2019], eBay [Pittman et al., 2017], Uber [Hamad et al., 2018]), social networks (e.g., Facebook [Noy et al., 2019], LinkedIn [He et al., 2016]), finance (e.g., Accenture [Okorafor and Ray, 2019], Banca d’Italia [Bellomarini et al., 2019], Bloomberg [Meij, 2019], Capital One [Branum and Sehon, 2019], Wells Fargo [Newman, 2019]), among others. Applications include search [Shrivastava, 2017, Singhal, 2012], recommendations [Chang, 2018, Hamad et al., 2018, He et al., 2016, Noy et al., 2019], personal agents [Pittman et al., 2017], advertising [He et al., 2016], business analytics [He et al., 2016], risk assessment [Tobin, 2017, Dalgliesh, 2016], automation [Henson et al., 2019], and more besides. We will provide more details on the use of knowledge graphs in practice in Chapter 10.

Running example. To keep the discussion accessible, throughout the book, we present concrete examples in the context of a hypothetical knowledge graph relating to tourism in Chile (loosely inspired by related use-cases [Kärle et al., 2018, Lu et al., 2016]). The knowledge graph is managed by a tourism board that aims to increase tourism in the country and promote new attractions in strategic areas. The knowledge graph itself will eventually describe tourist attractions, cultural events, services, businesses, travel routes, etc. Some applications the organisation envisages are to:

Outline. The remainder of the book is structured as follows:

Chapter 2
outlines graph data models and the languages used to query them.
Chapter 3
describes representations of schema, identity, and context for graphs.
Chapter 4
presents deductive formalisms for representing and entailing knowledge.
Chapter 5
describes inductive techniques for learning from graphs.
Chapter 6
discusses the creation and enrichment of knowledge graphs.
Chapter 7
enumerates dimensions for assessing knowledge graph quality.
Chapter 8
discusses various techniques for knowledge graph refinement.
Chapter 9
introduces principles and protocols for publishing knowledge graphs.
Chapter 10
surveys some prominent knowledge graphs and their applications.
Chapter 11
concludes with future directions for knowledge graphs.
Appendix A
outlines the historical background for knowledge graphs.

Data Graphs

At the foundation of any knowledge graph is the principle of first applying a graph abstraction to data, resulting in an initial data graph. We now discuss a selection of graph-structured data models that are commonly used in practice to represent data graphs. We then discuss the primitives that form the basis of graph query languages used to interrogate such data graphs.

Models

Leaving aside graphs, let us assume that the tourism board from our running example has not yet decided how to model relevant data about attractions, events, services, etc. The board first considers using a tabular structure – in particular, relational databases – to represent the required data, and though they do not know precisely what data they will need to capture, they begin to design an initial relational schema. They begin with an Event table with five columns:

Event(name, venue, type, start, end)

where name and start together form the primary key of the table in order to uniquely identify recurring events. But as they start to populate the data, they encounter various issues: events may have multiple names (e.g., in different languages), events may have multiple venues, they may not yet know the start and end date-times for future events, events may have multiple types, and so forth. Incrementally addressing these modelling issues as the data become more diverse, they generate internal identifiers for events and adapt their relational schema until they have:

EventName(id,name), EventStart(id,start), EventEnd(id,end), EventVenue(id,venue), EventType(id,type)(2.1)

With the above schema, the organisation can now model events with \(0{-}n\) names, venues, and types, and \(0{-}1\) start dates and end dates (without needing relational nulls).

Along the way, the board has to incrementally change the schema several times in order to support new sources of data. Each such change requires a costly remodelling, reloading, and reindexing of data; here we only considered one table. The tourism board struggles with the relational model because they do not know, a priori, what data will need to be modelled or what sources they will use. But once they reach the latter relational schema, the board finds that they can integrate further sources without more changes: with minimal assumptions on multiplicities (\(1{-}1\), \(1{-}n\), etc.) this schema offers a lot of flexibility for integrating incomplete and diverse data.

In fact, the refined, flexible schema that the board ends up with – as shown in (2.1) – is modelling a set of binary relations between entities, which indeed can be viewed as modelling a graph. By instead adopting a graph data model from the outset, the board could forgo the need for an upfront schema, and could define any (binary) relation between any pair of entities at any time.

We now introduce graph data models popular in practice [Angles et al., 2017].

Directed edge-labelled graphs

A directed edge-labelled graph (sometimes known as a multi-relational graph [Nickel and Tresp, 2013, Bordes et al., 2013, Balazevic et al., 2019a]) is defined as a set of nodes – like Santiago, Arica, EID16, 2018-03-22 12:00 – and a set of directed labelled edges between those nodes, like Santa LucíacitySantiago. In the case of knowledge graphs, nodes are used to represent entities and edges are used to represent (binary) relations between those entities. Figure 2.1 provides an example of how the tourism board could model some relevant event data as a directed edge-labelled graph. The graph includes data about the names, types, start and end date-times, and venues for events.1note 1 We represent bidirectional edges as Viña del MarbusArica, which more concisely depicts two directed edges: Viña del MarbusArica and Viña del MarbusArica. Also while some naming conventions recommend more complete edge labels that include a verb, such as has venue or is valid from, in this book, for presentation purposes, we will omit the “has” and “is” verbs from such labels, using simply venue or valid from. Adding information to such a graph typically involves adding new nodes and edges (with some exceptions discussed later). Representing incomplete information requires simply omitting a particular edge; for example, the graph does not yet define a start/end date-time for the Food Truck festival.

Directed edge-labelled graph describing events and their venues
Directed edge-labelled graph describing events and their venues

Modelling data as a graph in this way offers more flexibility for integrating new sources of data, compared to the standard relational model, where a schema must be defined upfront and followed at each step. While other structured data models such as trees (XML, JSON, etc.) would offer similar flexibility, graphs do not require organising the data hierarchically (should venue be a parent, child, or sibling of type for example?). They also allow cycles to be represented and queried (e.g., note the directed cycle in the routes between Santiago, Arica, and Viña del Mar).

A standardised data model based on directed edge-labelled graphs is the Resource Description Framework (RDF) [Cyganiak et al., 2014], which has been recommended by the W3C. The RDF model defines different types of nodes, including Internationalized Resource Identifiers (IRIs) [Dürst and Suignard, 2005] which allow for global identification of entities on the Web; literals, which allow for representing strings (with or without language tags) and other datatype values (integers, dates, etc.); and blank nodes, which are anonymous nodes that are not assigned an identifier (for example, rather than create internal identifiers like EID15, EID16, in RDF, we have the option to use blank nodes). We will discuss these different types of nodes further in Section 3.2 when we speak about issues relating to identity.

We now formally define a directed edge-labelled graph, where we denote by \(\con\) a countably infinite set of constants.

Directed edge-labelled graph
A directed edge-labelled graph is a tuple \(G = (V,E,L)\), where \(V \subseteq \con\) is a set of nodes, \(L \subseteq \con\) is a set of edge labels, and \(E \subseteq V \times L \times V\) is a set of edges.

In reference to Figure 2.1, the set of nodes \(V\) has 15 elements, including Arica, EID16, etc. The set of edges \(E\) has 23 triples, including (Arica, flight, Santiago). Bidirectional edges are represented with two edges. The set of edge labels \(L\) has 8 elements, including start, flight, etc.

Definition 2.1 does not state that \(V\) and \(L\) are disjoint: though not present in the example, a node can also serve as an edge-label. The definition also permits that nodes and edge labels can be present without any associated edge. Either restriction could be explicitly stated – if necessary – in a particular application while still conforming to a directed edge-labelled graph.

For ease of presentation presentation, we may treat a set of (directed labelled) edges \(E \subseteq V \times L \times V\) as a directed edge-labelled graph \((V,E,L)\), in which case we refer to the graph induced by \(E\) assuming that \(V\) and \(L\) contain all and only those nodes and edge labels, respectively, used in \(E\). We may similarly apply set operators on directed edge-labelled graphs, which should be interpreted as applying to their sets of edges; for example, given \(G_1 = (V_1,E_1,L_1)\) and \(G_2 = (V_2,E_2,L_2)\), by \(G_1 \cup G_2\) we refer to the directed edge-labelled graph induced by \(E_1 \cup E_2\).

Heterogeneous graphs

A heterogeneous graph [Hussein et al., 2018, Wang et al., 2019, Yang et al., 2020] (or heterogeneous information network [Sun et al., 2011, Sun and Han, 2012]) is a directed graph where each node and edge is assigned one type. Heterogeneous graphs are thus akin to directed edge-labelled graphs – with edge labels corresponding to edge types – but where the type of node forms part of the graph model itself, rather than being expressed with a relation (as seen in Figure 2.2). An edge is called homogeneous if it is between two nodes of the same type (e.g., borders in Figure 2.2); otherwise it is called heterogeneous (e.g., capital in Figure 2.2). Heterogeneous graphs allow for partitioning nodes according to their type, for example, for the purposes of machine learning tasks [Hussein et al., 2018, Wang et al., 2019, Yang et al., 2020]. Conversely, such graphs typically only support a one-to-one relation between nodes and types, which is not the case for directed edge-labelled graphs (see, for example, the node Santiago with zero types and EID15 with multiple types in Figure 2.1.

Del graph
Directed edge-labelled graph
Heterogenous graph
Heterogenous graph
Comparing directed edge-labelled graphs and heterogeneous graphs

We next define the notion of a heterogeneous graph.

Heterogeneous graph
A heterogeneous graph is a tuple \(G = (V,E,L,l)\), where \(V \subseteq \con\) is a set of nodes, \(L \subseteq \con\) is a set of edge/node labels, \(E \subseteq V \times L \times V\) is a set of edges, and \(l : V \rightarrow L\) maps each node to a label.

In reference to Figure 2.2b, the set of nodes \(V\) has three elements: Santiago, Chile, and Perú. The set of edges \(E\) has 3 triples, including (Santiago, capital, Chile). The set of edge labels \(L\) has 4 elements: capital, borders, City, Country. Finally, with respect to the node labels, \(l(\)Santiago\() =\) City, \(l(\)Chile\() =\) Country, and \(l(\)Perú\() =\) Country.

In heterogeneous graphs, edge and node labels are often called types. By rather defining edges with labels as per directed edge-labelled graphs – rather than separately labelling edges with \(l\) – two nodes can be related by \(n\) edges with \(n\) different labels; for example, we can represent both \((\)Santiago, capital, Chile\()\) and \((\)Santiago, country, Chile\()\) as edges in the heterogeneous graph.

Property graphs

Property graphs constitute an alternative graph model that offers additional flexibility when modelling more complex relations. Consider integrating incoming data that provide further details on which companies offer fares on which flights, allowing the board to better understand available routes between cities (for example, on national airlines). In the case of directed edge-labelled graphs, we cannot directly annotate an edge like SantiagoflightArica with the company (or companies) offering that route. But we could add a new node denoting a flight, connect it with the source, destination, companies, and mode, as shown in Figure 2.3a. Applying this modelling to all routes in Figure 2.1 would, however, involve significant changes.

The property graph model was thus proposed to offer additional flexibility when modelling data as a graph [Miller, 2013, Angles et al., 2017]. A property graph allows a set of property–value pairs and a label to be associated with both nodes and edges. Figure 2.3b depicts an example of a property graph with data analogous to Figure 2.3a. We use property–value pairs on edges to model the companies. The type of relation is captured by the label flight. We further use node labels to indicate the types of the two nodes, and property–value pairs for their latitude and longitude.

Directed edge-labelled graph
Directed edge-labelled graph
Property graph
Property graph
Comparing directed edge-labelled graphs and property graphs

Property graphs are prominently used in graph databases, such as Neo4j [Miller, 2013, Angles et al., 2017]. Property graphs can be converted to/from directed edge-labelled graphs [Hernández et al., 2015, Angles et al., 2019] (per, e.g., Figure 2.3b). In summary, directed edge-labelled graphs offer a more minimal model, while property graphs offer a more flexible one. Often the choice of model will be secondary to other practical factors, such as the implementations available for different models, etc.

We formally define a property graph.

Property graph
A property graph is a tuple \(G = (V,E,L,P,U,e,l,p)\), where \(V \subseteq \con\) is a set of node ids, \(E \subseteq \con\) is a set of edge ids, \(L \subseteq \con\) is a set of labels, \(P \subseteq \con\) is a set of properties, \(U \subseteq \con\) is a set of values, \(e : E \rightarrow V \times V\) maps an edge id to a pair of node ids, \(l : V \cup E \rightarrow 2^L\) maps a node or edge id to a set of labels, and \(p : V \cup E \rightarrow 2^{P \times U}\) maps a node or edge id to a set of property–value pairs.

Returning to Figure 2.3b:

  • the set \(V\) contains Santiago and Arica;
  • the set \(E\) contains LA380 and LA381;
  • the set \(L\) contains Capital City, Port City, and flight;
  • the set \(P\) contains lat, long, and company;
  • the set \(U\) contains –33.45, –70.66, LATAM, –18.48, and –70.33;
  • the mapping \(e\) gives, for example, \(e(\)LA380\() = (\)Santiago, Arica\()\);
  • the mapping \(l\) gives, for example, \(l(\)Santiago\() =\{ \)Capital City\(\}\) and \(l(\)LA380\() =\)\(\{ \)flight\(\}\);
  • the mapping \(p\) gives, for example, \(p(\)LA380\() =\{ (\)company, LATAM\() \}\) and \(p(\)Santiago\() =\)\(\{ (\)lat, –33.45\(), (\)long, –70.66\() \}\).

Unlike previous definitions [Angles et al., 2017], we allow a node or edge to have several values for a given property. In practice, systems like Neo4j [Miller, 2013] may rather support this by allowing a single array (i.e., list) of values.

Graph dataset

Although multiple directed edge-labelled graphs can be merged by taking their union, it is often desirable to manage several graphs rather than one monolithic graph; for example, it may be beneficial to manage multiple graphs from different sources, making it possible to update or refine data from one source, to distinguish untrustworthy sources from more trustworthy ones, and so forth. A graph dataset then consists of a set of named graphs and a default graph. Each named graph is a pair of a graph ID and a graph. The default graph is a graph without an ID, and is referenced “by default” if a graph ID is not specified. Figure 2.4 provides an example where events and routes are stored in two named graphs, and the default graph manages metadata about the named graphs. Graph names can also be used as nodes in a graph. Furthermore, nodes and edges can be repeated across graphs, where the same node in different graphs will typically refer to the same entity, allowing data on that entity to be integrated when merging graphs. Though the example depicts a dataset of directed edge-labelled graphs, the concept generalises straightforwardly to datasets of other types of graphs.

Graph dataset with two named graphs and a default graph describing events and routes
Graph dataset based on directed edge-labelled graphs with two named graphs and a default graph describing events and routes

An RDF dataset is a graph dataset model standardised by the W3C [Cyganiak et al., 2014] where each graph is an RDF graph, and graph names can be blank nodes or IRIs. A prominent use-case for RDF datasets is to manage and query Linked Data composed of interlinked documents of RDF graphs spanning the Web. When dealing with Web data, tracking the source of data becomes of key importance [Dividino et al., 2009, Bonatti et al., 2011, Zimmermann et al., 2012]. We will discuss Linked Data later in Section 3.2 and further discuss provenance in Section 3.3.

We more formally define a graph dataset. We assume that all data graphs featured in a given graph dataset follow the same model (directed edge-labelled graph, heterogeneous graph, property graph, etc).

Graph dataset
A named graph is a pair \((n,G)\) where \(G\) is a data graph, and \(n \in \con\) is a graph name. A graph dataset is a pair \(D = (G_D,N)\) where \(G_D\) is a data graph called the default graph and \(N\) is either the empty set, or a set of named graphs \(\{ (n_1,G_1), \ldots (n_k,G_k) \}\) (\(k > 0\)) such that if \(i \neq j\) then \(n_i \neq n_j\) (for all \(1 \leq i \leq k\), \(1 \leq j \leq k\)).

Figure 2.4 provides an example of a directed edge-labelled graph dataset \(D\) consisting of two named graphs and a default graph. The default graph does not have a name associated with it. The two graph names are Events and Routes; these are also used as nodes in the default graph.

Other graph data models

The previous models are popular examples of graph representations. Other graph data models exist with complex nodes that may contain individual edges [Angles and Gutierrez, 2008, Hartig and Thompson, 2014] or nested graphs [Angles and Gutierrez, 2008, Berners-Lee and Connolly, 2011] (sometimes called hypernodes [Levene and Poulovassilis, 1989]. Likewise the mathematical notion of a hypergraph defines complex edges that connect sets rather than pairs of nodes. In our view, a knowledge graph can adopt any such graph data model based on nodes and edges: often data can be converted from one model to another (see Figure 2.3a vs. Figure 2.3b). In the rest of the paper, we prefer discussing directed edge-labelled graphs given their relative succinctness, but most discussion extends naturally to other models.

Graph stores

A variety of techniques have been proposed for storing and indexing graphs, facilitating the efficient evaluation of queries (as discussed next). Directed edge-labelled graphs can be stored in relational databases either as a single relation of arity three (triple table), as a binary relation for each property (vertical partitioning), or as \(n\)-ary relations for entities of a given type (property tables) [Wylot et al., 2018]. Custom (so-called native) storage techniques have also been developed for a variety of graph models, providing efficient access for finding nodes, edges and their adjacent elements [Angles and Gutierrez, 2008, Miller, 2013, Wylot et al., 2018]. A number of systems further allow for distributing graphs over multiple machines based on popular NoSQL stores or custom partitioning schemes [Wylot et al., 2018, Janke and Staab, 2018]. For further details we refer to the book chapter by Janke and Staab [2018] and the survey by Wylot et al. [2018] dedicated to this topic.

Querying

A number of languages have been proposed for querying graphs [Angles et al., 2017], including the SPARQL query language for RDF graphs [Harris et al., 2013]; and Cypher [Francis et al., 2018], Gremlin [Rodriguez, 2015], and G-CORE [Angles et al., 2018] for querying property graphs. We refer to Seifer et al. [2019] for an investigation of the popularity of these languages. Underlying these query languages are some common primitives, including (basic) graph patterns, relational operators, path expressions, and more besides [Angles et al., 2017]. We now describe these core features for querying graphs in turn, starting with basic graph patterns.

Basic graph patterns

At the core of every structured query language for graphs lie basic graph patterns [Consens and Mendelzon, 1990, Angles et al., 2017], which follow the same model as the data graph being queried (see Section 2.1), additionally allowing variables as terms.2note 2 The terms of a directed edge-labelled graph are its nodes and edge-labels. The terms of a property graph are its ids, labels, properties, and values (as used on either edges or nodes). Terms in basic graph patterns are thus divided into constants, such as Arica or venue, and variables, which we prefix with question marks, such as ?event or ?rel. A basic graph pattern is then evaluated against the data graph by generating mappings from the variables of the graph pattern to constants in the data graph such that the image of the graph pattern under the mapping (replacing variables with the assigned constants) is contained within the data graph.

Figure 2.5 provide an example of a basic graph pattern looking for the venues of Food Festivals, along with the possible mappings generated by the graph pattern against the data graph of Figure 2.1. In some of the presented mappings (the last two listed), multiple variables are mapped to the same term, which may or may not be desirable depending on the application. Hence a number of semantics have been proposed for evaluating basic graph patterns [Angles et al., 2017], amongst which the most important are: homomorphism-based semantics, which allows multiple variables to be mapped to the same term such that all mappings shown in Figure 2.5 would be considered results; and isomorphism-based semantics, which requires variables on nodes and/or edges to be mapped to unique terms, thus excluding the latter three mappings of Figure 2.5 from the results. Different languages may adopt different semantics for evaluating basic graph patterns; for example, SPARQL adopts a homomorphism-based semantics, while Cypher adopts an isomorphism-based semantics specifically on edges (while allowing multiple variables to map to one node).

Graph pattern
 
?ev ?vn1 ?vn2
EID16 Piscina Olímpica Sotomayor
EID16 Sotomayor Piscina Olímpica
EID16 Piscina Olímpica Piscina Olímpica
EID16 Sotomayor Sotomayor
EID15 Santa Lucía Santa Lucía
 
basic directed edge-labelled graph pattern (left) with mappings generated over the directed edge-labelled graph of Figure 2.1 (right)

As we will see in later examples (particularly Figure 2.7), basic graph patterns may also form cycles (be they directed or undirected), and may replace edge labels with variables. Basic graph patterns in the context of other models – such as property graphs – can be defined analogously by allowing variables to replace constants in any position of the model.

We formalise basic graph patterns first for directed edge-labelled graphs, and subsequently for property graphs [Angles et al., 2017]. For these definitions, we introduce a countably infinite set of variables \(\var\) ranging over (but disjoint from: \(\con \cap \var = \emptyset\)) the set of constants. We refer generically to constants and variables as terms, denoted and defined as \(\term = \con \cup \var\). We define a basic graph pattern for a model by simply replacing constants with terms (that may be variables). Though we focus on directed edge-labelled graphs and property graphs, basic graph patterns for other graph models can be defined analogously.

Basic directed edge-labelled graph pattern
We define a basic directed edge-labelled graph pattern as a tuple \(Q = (V,E,L)\), where \(V \subseteq \term\) is a set of node terms, \(L \subseteq \term\) is a set of edge terms, and \(E \subseteq V \times L \times V\) is a set of edges (triple patterns).

Returning to the example of Figure 2.5:

  • the set \(V\) contains the constant Food Festival and variables ?event, ?ven1 and ?ven2;
  • the set \(E\) contains four edges, including \((\)?event, type, Food Festival\()\);
  • the set \(L\) contains the constants type and venue.

A basic property graph pattern is also defined by introducing variables.

Basic property graph pattern
We define a basic property graph pattern as a tuple \(Q = (V,E,L,P,U,e,l,p)\), where \(V \subseteq \term\) is a set of node id terms, \(E \subseteq \term\) is a set of edge id terms, \(L \subseteq \term\) is a set of label terms, \(P \subseteq \term\) is a set of property terms, \(U \subseteq \term\) is a set of value terms, \(e : E \rightarrow V \times V\) maps an edge id term to a pair of node id terms, \(l : V \cup E \rightarrow 2^{L}\) maps a node or edge id term to a set of label terms, and \(p : V \cup E \rightarrow 2^{P \times U}\) maps a node or edge id term to a set of pairs of property–value terms.

Towards defining the results of evaluating a basic graph pattern over a data graph (following the same model), we first define a partial mapping \(\mu : \var \rightarrow \con\) from variables to constants, whose domain (the set of variables for which it is defined) is denoted by \(\dom(\mu)\). Given a basic graph pattern \(Q\), let \(\var(Q)\) denote the set of all variables appearing in (some recursively nested element of) \(Q\). We further denote by \(\mu(Q)\) the image of \(Q\) under \(\mu\), meaning that any variable \(v \in \var(Q) \cap \dom(\mu)\) is replaced in \(Q\) by \(\mu(v)\). Observe that when \(\var(Q) \subseteq \dom(\mu)\), then \(\mu(Q)\) is a data graph (in the corresponding model of \(Q\)).

Next, we define the notion of containment between data graphs. For two directed edge-labelled graphs \(G_1 = (V_1,E_1,L_1)\) and \(G_2 = (V_2,E_2,L_2)\), we say that \(G_1\) is a sub-graph of \(G_2\), denoted \(G_1 \subseteq G_2\), if and only if \(V_1 \subseteq V_2\), \(E_1 \subseteq E_2\), and \(L_1 \subseteq L_2\).3note 3 Given, for example, \(G_1 = (\{a\},\{(a,b,a)\},\{b,c\})\) and \(G_2 = (\{a,c\},\{(a,b,a)\},\{b\})\), we remark that \(G_1 \not\subseteq G_2\) and \(G_2 \not\subseteq G_1\): the former has a label not used on an edge while the latter has a node without an incident edge. In concrete data models like RDF where such cases of nodes or labels without edges cannot occur, the sub-graph relation \(G_1 \subseteq G_2\) holds if and only if \(E_1 \subseteq E_2\) holds. Conversely, in property graphs, nodes can often be defined without edges. For two property graphs \(G_1 = (V_1,E_1,L_1,P_1,U_1,e_1,l_1,p_1)\) and \(G_2 = (V_2,E_2,L_2,P_2,U_2,e_2,l_2,p_2)\), we say that \(G_1\) is a sub-graph of \(G_2\), denoted \(G_1 \subseteq G_2\), if and only if \(V_1 \subseteq V_2\), \(E_1 \subseteq E_2\), \(L_1 \subseteq L_2\), \(P_1 \subseteq P_2\), \(U_1 \subseteq U_2\), for all \(x \in E_1\) it holds that \(e_1(x) = e_2(x)\), and for all \(y \in E_1 \cup V_1\) it holds that \(l_1(y) \subseteq l_2(y)\) and \(p_1(y) \subseteq p_2(y)\).

We are now ready to define the evaluation of a basic graph pattern.

Evaluation of a basic graph pattern
Let \(Q\) be a basic graph pattern and let \(G\) be a data graph (in the same model). We then define the evaluation of the basic graph pattern \(Q\) over the data graph \(G\), denoted \(Q(G)\), to be the set of mappings \(Q(G) = \{ \mu \mid \mu(Q) \subseteq G \text{ and } \dom(\mu) = \var(Q) \}\).

Figure 2.5 enumerates all of the mappings given by the evaluation of the depicted basic graph pattern over the data graph of Figure 2.1. Each non-header row indicates a mapping \(\mu\).

The final results of evaluating a basic graph pattern may vary depending on the choice of semantics: the results under homomorphism-based semantics are defined as \(Q(G)\). Conversely, under isomorphism-based semantics, mappings that send two edge variables to the same constant and/or mappings that send two node variables to the same constant may be excluded from the results. Henceforth we assume the more general homomorphism-based semantics.

Complex graph patterns

A (basic) graph pattern transforms an input graph into a table of results (as shown in Figure 2.5). We may then consider using the relational algebra to combine and/or transform such tables, thus forming more complex queries from one or more graph patterns. Recall that the relational algebra consists of unary operators that accept one input table, and binary operators that accept two input tables. Unary operators include projection (\(\pi\)) to output a subset of columns, selection (\(\sigma\)) to output a subset of rows matching a given condition, and renaming of columns (\(\rho\)). Binary operators include union (\(\cup\)) to merge the rows of two tables into one table, difference (\(-\)) to remove the rows from the first table present in the second table, and joins (\(\Join\)) to extend the rows of one table with rows from the other table that satisfy a join condition. Selection and join conditions typically include equalities (\(=\)), inequalities (\(\leq\)), negation (\(\neg\)), disjunction (\(\vee\)), etc. From these operators, we can further define other (syntactic) operators, such as intersection (\(\cap\)) to output rows in both tables, anti-join (\(\rhd\), aka minus) to output rows from the first table for which there are no join-compatible rows in the second table, left-join (⟕, aka optional) to perform a join but keeping rows from the first table without a compatible row in the second table, etc.

Basic graph patterns can then be expressed in a subset of relational algebra (namely \(\pi\), \(\sigma\), \(\rho\), \(\Join\)). Assuming, for example, a single ternary relation \(G(s,p,o)\) representing a graph – i.e., a table \(G\) with three columns \(s\), \(p\), \(o\) – the query of Figure 2.5 can be expressed in relational algebra as:

\(\pi_{ev,vn_1,vn_2}(\sigma_{p=\texttt{type} \wedge o=\texttt{Food Festival} \wedge p_1=p_2=\texttt{venue}}(\rho_{s/ev}(G \bowtie \rho_{p/p_1,o/vn_1}(G) \bowtie \rho_{p/p_2,o/vn_2}(G))))\)

where \(\Join\) denotes a natural join, meaning that equality is checked across pairs of columns with the same name in both tables (here, the join is thus performed on the subject column \(s\)). The result of this query is a table with a column for each variable: \(ev,vn1,vn2\). However, not all queries using \(\pi, \sigma, \rho\) and \(\Join\) on \(G\) can be expressed as basic graph patterns; for example, we cannot choose which variables to project in a basic graph pattern, but rather must project all variables not fixed to a constant.

Graph query languages such as SPARQL [Harris et al., 2013] and Cypher [Francis et al., 2018] allow the full use of relational operators over the results of graph patterns, giving rise to complex graph patterns [Angles et al., 2017]. Figure 2.6 presents an example of a complex graph pattern with projected variables in bold, choosing particular variables to appear in the final results. In Figure 2.7, we give another example of a complex graph pattern looking for food festivals or drinks festivals not held in Santiago, optionally returning their start date and name (where available).

Conjunctive query
?name1 ?con ?name2
Food Truck bus Food Truck
Food Truck bus Food Truck
Food Truck bus Ñam
Food Truck flight Ñam
Food Truck flight Ñam
Ñam bus Food Truck
Ñam flight Food Truck
Ñam flight Food Truck
Complex graph pattern (left) with mappings generated over the graph of Figure 2.1 (right)

Complex graph patterns can give rise to duplicate results; for example, the first result in Figure 2.6 appears twice since ?city1 matches Arica and ?city2 matches Viña del Mar in one result, and vice-versa in the other. Query languages then offer two semantics: bag semantics preserves duplicates according to the multiplicity of the underlying mappings, while set semantics (typically invoked with a DISTINCT keyword) removes duplicates from the results.

Complex graph pattern 1 Complex graph pattern 2 Complex graph pattern 3 Complex graph pattern 4 Complex graph pattern 5
\(Q := ((((Q_1 \cup Q_2) \rhd Q_3)\) ⟕ \(Q_4 )\) ⟕ \(Q_5),\qquad Q(G) =\)
?event?start?name
EID16Food Truck
Complex graph pattern (\(Q\)) with mappings generated (\(Q(G)\)) over the graph of Figure 2.1 (\(G\))

We now formally define complex graph patterns.

Complex graph pattern
Complex graph patterns are defined recursively, as follows:
  • If \(Q\) is a basic graph pattern, then \(Q\) is a complex graph pattern.
  • If \(Q\) is a complex graph pattern, and \(\mathcal{V} \subseteq \var(Q)\), then \(\pi_\mathcal{V}(Q)\) is a complex graph pattern.
  • If \(Q\) is a complex graph pattern, and \(R\) is a selection condition with Boolean and equality connectives (\(\wedge\), \(\vee\), \(\neg\), \(=\)) , then \(\sigma_R(Q)\) is a complex graph pattern.
  • If both \(Q_1\) and \(Q_2\) are complex graph patterns, then \(Q_1 \Join Q_2\), \(Q_1 \cup Q_2\), \(Q_1 - Q_2\) and \(Q_1 \rhd Q_2\) are also complex graph patterns.

We now define the evaluation of complex graph patterns. Given a mapping \(\mu\), for a set of variables \(\mathcal{V} \subseteq \var\) let \(\mu[\mathcal{V}]\) denote the mapping \(\mu'\) such that \(\dom(\mu') = \dom(\mu) \cap \mathcal{V}\) and \(\mu'(v) = \mu(v)\) for all \(v \in \dom(\mu')\) (in other words, \(\mu[\mathcal{V}]\) projects the variables \(\mathcal{V}\) from \(\mu\)). Letting \(R\) denote a Boolean selection condition and \(\mu\) a mapping, we denote by \(\mu \models R\) that \(\mu\) satisfies the Boolean condition. Finally, we define two mappings \(\mu_1\) and \(\mu_2\) to be compatible, denoted \(\mu_1 \sim \mu_2\), if and only if \(\mu_1(v) = \mu_2(v)\) for all \(v \in \dom(\mu_1) \cap \dom(\mu_2)\) (i.e., they map common variables to the same constant). We are now ready to provide the definition.

Complex graph pattern evaluation
Given a complex graph pattern \(Q\), if \(Q\) is a basic graph pattern, then \(Q(G)\) is defined per Definition 2.7. Otherwise, \(Q(G)\) is defined as follows: \begin{align*} \pi_\mathcal{V}(Q)(G) = & \,\{ \mu[\mathcal{V}] \mid \mu \in Q(G) \} \\ \sigma_R(Q)(G) = & \, \{ \mu \mid \mu \in Q(G)\text{ and }\mu \models R\}\\ Q_1 \Join Q_2(G) = & \,\{ \mu_1 \cup \mu_2 \mid \mu_1 \in Q_2(G), \mu_2 \in Q_1(G)\text{ and }\mu_1 \sim \mu_2 \} \\ Q_1 \cup Q_2(G) = & \,\{ \mu \mid \mu \in Q_1(G)\text{ or } \mu \in Q_2(G) \} \\ Q_1 - Q_2(G) = & \,\{ \mu \mid \mu \in Q_1(G)\text{ and } \mu \notin Q_2(G) \} \\ Q_1 \rhd Q_2(G) = & \,\{ \mu \mid \mu \in Q_1(G)\text{ and }\nexists \mu_2 \in Q_2(G)\text{ such that }\mu \sim \mu_2 \} \end{align*}

Based on these operators, we can define some additional syntactic operators, such as the left-join ⟕, aka optional):

\begin{align*} Q_1 ⟕ Q_2(G) = & \,(Q_1(G) \Join Q_2(G)) \cup (Q_1(G) \rhd Q_2(G)) \end{align*}

We call such operators syntactic as they do not add expressivity.

Figure 2.7 illustrates a complex graph pattern and its evaluation.

Navigational graph patterns

A key feature that distinguishes graph query languages is the ability to include path expressions in queries. A path expression \(r\) is a regular expression that allows for matching arbitrary-length paths between two nodes using a regular path query \((x,r,y)\), where \(x\) and \(y\) can be variables or constants (or even the same term). The base path expression is where \(r\) is a constant (an edge label). Furthermore if \(r\) is a path expression, then \(r^*\) (Kleene star: zero-or-more) is also a path expression. Finally, if \(r_1\) and \(r_2\) are path expressions, then \(r_1 \mid r_2\) (disjunction) and \(r_1 \cdot r_2\) (concatenation) are also path expressions. A related notion is that of 2-way regular path queries, which also allow for querying inverse paths; specifically, if \(r\) is path expression, then it is a 2-way path expression, and if \(r\) is a 2-way path expression, then \(r^-\) (inverse) is a 2-way path expression. Henceforth we will refer generically to both the 1-way and 2-way variants as path expressions and regular path queries.

Regular path queries can be evaluated under a number of different semantics. For example, \((\)Arica, bus*, ?city\()\) evaluated against the graph of Figure 2.1 may match the paths shown in Figure 2.8. In fact, since a cycle is present, an infinite number of paths are potentially matched. For this reason, restricted semantics are often applied, returning only the shortest paths, or paths without repeated nodes or edges (as in the case of Cypher).4note 4 Mapping variables to paths requires special treatment [Angles et al., 2017]. Cypher [Francis et al., 2018] returns a string that encodes a path, upon which certain functions such as length(·) can be applied. G-CORE [Angles et al., 2018], on the other hand, allows for returning paths, and supports additional operators on them, including projecting them as graphs, applying cost functions, and more besides. Rather than returning paths, another option is to instead return the (finite) set of pairs of nodes connected by a matching path (as in the case of SPARQL 1.1).

Path matching 1 Path matching 2 Path matching 3 Path matching 4
Example paths matching \((\)Arica, bus*, ?city\()\) over the graph of Figure 2.1

Regular path queries can then be used in basic graph patterns to express navigational graph patterns [Angles et al., 2017], as shown in Figure 2.9, which illustrates a query searching for food festivals in cities reachable (recursively) from Arica by bus or flight. Furthermore, when regular path queries and graph patterns are combined with operators such as projection, selection, union, difference, and optional, the result is known as complex navigational graph patterns [Angles et al., 2017].

We first define path expressions and regular path queries.

Path expression
A constant (edge label) \(c\) is a path expression. Furthermore, if \(r\), \(r_1\) and \(r_2\) are path expressions, then:
  • \(r^-\) (inverse) and \(r^*\) (Kleene star) are path expressions.
  • \(r_1 \cdot r_2\) (concatenation) and \(r_1 \mid r_2\) (disjunction) are path expressions.

We now define the evaluation of a path expression on a directed-edge labelled graph under the SPARQL 1.1-style semantics whereby the endpoints (pairs of start and end nodes) of the path are returned [Harris et al., 2013].

Path evaluation (directed edge-labelled graph)
Given a directed edge-labelled graph \(G = (V,E,L)\) and a path expression \(r\), we define the evaluation of \(r\) over \(G\), denoted \(r[G]\), as follows: \begin{align*} r[G] = &\, \{ (u,v) \mid (u,r,v) \in E \} \,(\text{for }r \in \con) \\ r^-[G] = &\, \{ (u,v) \mid (v,u) \in r[G] \} \\ r_1 \mid r_2[G] = &\, r_1[G] \cup r_2[G] \\ r_1 \cdot r_2[G] = &\, \{ (u,v) \mid \exists w \in V : (u,w) \in r_1[G]\text{ and }(w,v) \in r_2[G]\}\\ r^*[G] = &\, V \cup \bigcup_{n \in \mathbb{N^+}} r^n[G] \end{align*} where by \(r^n\) we denote the \(n\)th-concatenation of \(r\) (e.g., \(r^3 = r \cdot r \cdot r\)).

The evaluation of a path expression on a property graph \(G = (V,E,L,P,U,e,l,p)\) can be defined analogously by adapting the first definition (in the case that \(r \in \con\)) as follows:

\[ r[G] = \{(u,v) \mid \exists x \in E : e(x) = (u,v)\text{ and }l(e) = r \} \,.\]

The rest of the definitions then remain unchanged.

Query languages may support additional operators, some of which are syntactic (e.g., \(r^+\) is sometimes used for one-or-more, but can be rewritten as \(r \cdot r^*\)), while others may add expressivity such as the case of SPARQL [Harris et al., 2013], which allows a limited form of negation in expressions (e.g., \(!r\), with \(r\) being a constant or the inverse of a constant, matching any path not labelled \(r\)).

Next we define a regular path query and its evaluation.

Regular path query
A regular path query is a triple \((x,r,y)\) where \(x,y \in \con \cup \var\) and \(r\) is a path expression.
Regular path query evaluation
Let \(G\) denote a directed edge-labelled graph, \(c\), \(c_1\), \(c_2 \in \con\) denote constants and \(z\), \(z_1\), \(z_2 \in \var\) denote variables. Then the evaluation of a regular path query is defined as follows: \begin{align*} (c_1,r,c_2)(G) = & \{ \mu_\emptyset \mid (c_1,c_2) \in r[G] \} \\ (c,r,z)(G) = & \{ \mu \mid \dom(\mu) = \{ z \}\text{ and }(c,\mu(z)) \in r[G] \} \\ (z,r,c)(G) = & \{ \mu \mid \dom(\mu) = \{ z \}\text{ and }(\mu(z),c) \in r[G] \} \\ (z_1,r,z_2)(G) = & \{ \mu \mid \dom(\mu) = \{ z_1, z_2 \}\text{ and }(\mu(z_1),\mu(z_2)) \in r[G] \} \end{align*} where \(\mu_\emptyset\) denotes the empty mapping such that \(\dom(\mu) = \emptyset\) (the join identity).
Navigational graph pattern
If \(Q\) is a basic graph pattern, then \(Q\) is a navigational graph pattern. If \(Q\) is a navigational graph pattern and \((x,r,y)\) is a regular path query, then \(Q \Join (x,r,y)\) is a navigational graph pattern.

The definition of the evaluation of a navigational graph pattern then follows from the previous definition of a join and the definition of the evaluation of a regular path query (for a directed edge-labelled graph or a property graph, respectively). Likewise, complex navigational graph patterns – and their evaluation – are defined by extending this definition in the natural way with the same operators from Definition 2.8 following the same semantics seen in Definition 2.9.

Navigational graph pattern
 
?event ?name ?city
EID15 Ñam Santiago
EID16 Food Truck Arica
EID16 Food Truck Viña del Mar
 
Navigational graph pattern (left) with mappings generated over the graph of Figure 2.1 (right)

Other features

Thus far, we have discussed features that form the practical and theoretical foundation of any query language for graphs [Angles et al., 2017]. However, specific query languages for graphs may support other features, such as aggregation (GROUP BY, COUNT, etc.), more complex filters and datatype operators (e.g., range queries on years extracted from a date), federation for querying remotely hosted graphs over the Web, languages for updating graphs, support for entailment, etc. For more information, we refer to the documentation of the respective query languages (e.g., [Harris et al., 2013, Angles et al., 2018]) and to the survey by Angles et al. [2017].

Query Interfaces

Knowledge graphs are often queried by non-expert users who may not be able to express their information needs in terms of a particular graph query language. Different types of interfaces have thus been proposed in order to assist users in querying data graphs. Such interfaces may support, for example:

Faceted browsing:
Users start by specifying a simple search, such as a keyword search, a type of node like Food Festival, or possibly other kinds of search. They are then presented with a set of matching results, and a set of facets, which are typically attributes (e.g., venue) and values (e.g., Santa Lucía) present in the current results set. Selecting a value for a facet restricts the current results set to include only results with the indicated value; this selection process can be applied iteratively to restrict results per multiple facets. Often the faceted criteria are translated into and evaluated as graph queries. Though relatively intuitive for users, such systems typically support acyclic queries that generate lists of results (analogous to graph queries that project a single variable), and rarely support more expressive queries. Examples of faceted browsing systems for graphs include VisiNav [Harth, 2010], Broccoli [Bast and Buchhold, 2013], SemFacet [Arenas et al., 2016], GraFa [Moreno-Vega and Hogan, 2018], etc.
Query building:
Users are provided with a form or graphical interface that can be used to specify a graph query without needing to understand the syntax of a specific query language. Such query builders allow for incrementally adding nodes or edges to the query, assisted by features such as auto-completion, previewing intermediate results, and graph navigation. Query builders typically allow for expressing queries equivalent to (cyclic) basic graph patterns, but may not support more expressive features of query languages as described herein. Graph query builder systems include Smeagol [Clemmer and Davies, 2011], QueryVOWL [Haag et al., 2015], VIIQ [Jayaram et al., 2015a], Sparklis [Ferré, 2017], RDF Explorer [Vargas et al., 2019], and more besides.
Query-by-example:
Users provide examples of positive and sometimes negative answers to their queries. For example, they may provide as positive examples the nodes Arica, Santiago, Viña del Mar, and as negative examples the nodes Chile, Lima, where the system will then “reverse engineer” a query that returns positive examples but not negative examples (in this case, the query proposed may return nodes of type City whose country is Chile). Query-by-example systems typically support basic graph patterns, and may not support more expressive querying features. They are useful in cases where users have examples of what they are looking for, but are not necessarily sure of the query they need to retrieve similar examples. Query-by-example systems for graphs include GQBE [Jayaram et al., 2015b] and SPARQLByE [Diaz et al., 2016].
Question answering:
Users express their queries as questions in natural language; for example, they might ask “What food festivals will be held in Arica?”. The question answering system will then generate answers from the graph based on its best interpretation of the question. We identify three types of question answering system. Navigation-based systems identify entities/nodes from the graph that are mentioned in the query, and then attempt to navigate edges from those nodes whose labels best match the question; for example, they may match the nodes Food Festival and Arica in the graph based on the question, and from there, try to navigate edges in the graph whose labels match the question in order to find answers. Template-based systems rather pre-suppose a fixed list of question templates expressed in the query language, with placeholder variables that will be replaced with entities/nodes detected in the question; a template matched for the previous example may be of the form “What X will be held in Y?”. Translation-based systems attempt to translate the question into a query in the structured query language, using (typically neural) machine translation techniques. The latter two types of question answering systems can additionally return a graph query that explains the answers generated. Question answering systems are often very intuitive to use, but may not always return correct results, particularly when considering complex questions/queries. Examples of question answering systems for knowledge graphs include Treo [Freitas et al., 2011], NFF [Hu et al., 2018], TemplateQA [Zheng et al., 2018], WDAqua-core1 [Diefenbach et al., 2020], and more besides.

Such query interfaces enable non-expert users to formulate queries over graphs, which in turn broadens the potential impact of knowledge graphs.

Schema, Identity, Context

In this chapter we describe extensions of the data graph – relating to schema, identity and context – that provide additional structures for accumulating knowledge. Henceforth, we refer to a data graph as a collection of data represented as nodes and edges using one of the models discussed in Chapter 2. We refer to a knowledge graph as a data graph potentially enhanced with representations of schema, identity, context, ontologies and/or rules. These additional representations may be embedded in the data graph, or layered above. Representations for schema, identity and context are discussed now, while ontologies and rules will be discussed in Chapter 4.

Schema

One of the benefits of modelling data as graphs – versus, for example, the relational model – is the option to forgo or postpone the definition of a schema. However, when modelling data as graphs, schemata can be used to prescribe a high-level structure and/or semantics that the graph follows or should follow. We discuss three types of graph schemata: semantic, validating, and emergent.

Semantic schema

A semantic schema allows for defining the meaning of high-level terms (aka vocabulary or terminology) used in the graph, which facilitates reasoning over graphs using those terms. Looking at Figure 2.1, for example, we may notice some natural groupings of nodes based on the types of entities to which they refer. We may thus decide to define classes, such as Event, City, etc., to denote these groupings. In fact, Figure 2.1 already illustrates three low-level classes – Open Market, Food Market, Drinks Festival – grouping similar entities with an edge labelled type. We may subsequently wish to capture some relations between some of these classes. In Figure 3.1, we present a class hierarchy for events where children are defined to be sub-classes of their parents such that if we find an edge EID15typeFood Festival in our graph, we may also infer that EID15typeFestival and EID15typeEvent hold in the graph.

Example class hierarchy for Event
Example class hierarchy for Event

Aside from classes, we may also wish to define the semantics of edge labels, aka properties. Returning to Figure 2.1, we may consider that the properties city and venue are sub-properties of a more general property location, such that given an edge Santa LucíacitySantiago, for example, we may also infer that Santa LucíalocationSantiago must hold as an edge in the graph. We may also consider, for example, that bus and flight are both sub-properties of a more general property connects to. Along these lines, properties may also form a hierarchy similar to what we saw for classes. We may further define the domain of properties, indicating the class(es) of entities for nodes from which edges with that property extend; for example, we may define that the domain of connects to is a class Place, such that given the previous sub-property relations, we infer AricatypePlace. Conversely, we may define the range of properties, indicating the class(es) of entities for nodes to which edges with that property extend; for example, we may define that the range of city is a class City, inferring that AricatypeCity.

A prominent standard for defining a semantic schema for (RDF) graphs is the RDF Schema (RDFS) standard [Brickley and Guha, 2014], which allows for defining sub-classes, sub-properties, domains, and ranges amongst the classes and properties used in an RDF graph, where such definitions can be serialised as a graph. We illustrate the semantics of these features in Table 3.1 and provide a concrete example of definitions in Figure 3.2 for a sample of terms used in the running example. These definitions can then be embedded into a data graph. More generally, the semantics of terms used in a graph can be defined in much more depth than seen here, as is supported by the Web Ontology Language (OWL) standard [Hitzler et al., 2012] for RDF graphs. We will return to such semantics later in Chapter 4.

Definitions for sub-class, sub-property, domain and range
Feature Definition Condition Example
Sub-class \(c\)subc. of\(d\) \(x\)type\(c\) implies \(x\)type\(d\) Citysubc. ofPlace
Sub-property \(p\)subp. of\(q\) \(x\)\(p\)\(y\) implies \(x\)\(q\)\(y\) venuesubp. oflocation
Domain \(p\)domain\(c\) \(x\)\(p\)\(y\) implies \(x\)type\(c\) venuedomainEvent
Range \(p\)range\(c\) \(x\)\(p\)\(y\) implies \(y\)type\(c\) venuerangeVenue
Example schema graph describing sub-classes, sub-properties, domains, and ranges
Example schema with sub-classes, sub-properties, domains, and ranges

Semantic schemata are typically defined for incomplete graph data, where the absence of an edge between two nodes, such as Viña del MarflightArica, does not mean that the relation does not hold in the real world. Therefore, from the graph of Figure 2.1, we cannot assume that there is no flight between Viña del Mar and Arica. In contrast, if the Closed World Assumption (CWA) were adopted – as is the case in many classical database systems – it would be assumed that the data graph is a complete description of the world, thus allowing to assert with certainty that no flight exists between the two cities. Systems that do not adopt the CWA are said to adopt the Open World Assumption (OWA). Considering our running example, it would be unreasonable to assume that the tourism organisation has complete knowledge of everything describable in its knowledge graph, and hence adopting the OWA appears more appropriate. However, it can be inconvenient if a system is unable to definitely answer “yes” or “no” to questions such as “is there a flight between Arica and Viña del Mar?”, especially when the organisation is certain that it has complete knowledge of the flights. A compromise between OWA and CWA is the Local Closed World Assumption (LCWA), where portions of the data graph are assumed to be complete.

Validating schema

When graphs are used to represent diverse, incomplete data at large scale, the OWA is the most appropriate choice for a default semantics. But in some scenarios, we may wish to guarantee that our data graph – or specific parts thereof – are in some sense “complete”. Returning to Figure 2.1, for example, we may wish to ensure that all events have at least a name, a venue, a start date, and an end date, such that applications using the data – e.g., one that sends event notifications to users – can ensure that they have the minimal information required. Furthermore, we may wish to ensure that the city of an event is stated to be a city (rather than inferring that it is a city). We can define such constraints in a validating schema and validate the data graph with respect to the resulting schema, listing constraint violations (if any). Thus while semantic schemata allow for inferring new graph data, validating schemata allow for validating a given data graph with respect to some constraints.

A standard way to define a validating schema for graphs is using shapes [Knublauch and Kontokostas, 2017, Prud'hommeaux et al., 2014, Labra Gayo et al., 2018]. A shape targets a set of nodes in a data graph and specifies constraints on those nodes. The shape’s target can be defined in many ways, such as targeting all instances of a class, the domain or range of a property, the result of a query, nodes connected to the target of another shape by a given property, etc. Constraints can then be defined on the targeted nodes, such as to restrict the number or types of values taken on a given property, the shapes that such values must satisfy, etc.

A shapes graph is formed from a set of interrelated shapes. Shapes graphs can be depicted as UML-like class diagrams, where Figure 3.3 illustrates an example of a shapes graph based on Figure 2.1, defining constraints on four interrelated shapes. Each shape – denoted with a box like Place, Event, etc. – is associated with a set of constraints. Nodes conform to a shape if and only if they satisfy all constraints defined on the shape. Inside each shape box are placed constraints on the number (e.g., [1..*] denotes one-to-many, [1..1] denotes precisely one, etc.) and types (e.g., string, dateTime, etc.) of nodes that conforming nodes can relate to with a property (e.g., name, start, etc.). Another option is to place constraints on the number of nodes conforming to a particular shape that the conforming node can relate to with a property (thus generating edges between shapes); for example, Eventvenue
1..*
Venue denotes that conforming nodes for Event must relate to at least one node with the property venue that conforms to the Venue shape. Shapes can inherit the constraints of parent shapes – with inheritance denoted with an \(\triangle\) connector – as in the case of City and Venue, whose conforming nodes must also conform to the Place shape.

Example shapes graph depicted as a UML-like diagram
Example shapes graph depicted as a UML-like diagram

Given a shape and a targeted node, it is possible to check if the node conforms to that shape or not, which may require checking conformance of other nodes; for example, the node EID15 conforms to the Event shape not only based on its local properties, but also based on conformance of Santa Lucía to Venue and Santiago to City. Conformance dependencies may also be recursive, where the conformance of Santiago to City requires that it conforms to Place, which requires that Viña del Mar and Arica conform to Place, and so on. Conversely, EID16 does not conform to Event, as it does not have the start and end properties required by the example shapes graph.

When declaring shapes, the data modeller may not know in advance the entire set of properties that some nodes can have (now or in the future). An open shape allows the node to have additional properties not specified by the shape, while a closed shape does not. For example, if we add the edge SantiagofounderPedro de Valdivia to the graph represented in Figure 2.1, then Santiago only conforms to the City shape if the shape is defined as open (since the shape does not mention founder).

Practical languages for shapes often support additional Boolean features, such as conjunction (and), disjunction (or), and negation (not) of shapes; for example, we may say that all the values of venue should conform to the shape Venue and (not City), making explicit that venues in the data graph should not be directly given as cities. However, shapes languages that freely combine recursion and negation may lead to semantic problems, depending on how their semantics are defined. To illustrate, consider the following case inspired by the barber paradox [Labra Gayo et al., 2018], involving a shape Barber whose conforming nodes shave at least one node conforming to Person and (not Barber). Now, given (only) BobshaveBob with Bob conforming to Person, does Bob conform to Barber? If yes – if Bob conforms to Barber – then Bob violates the constraint by not shaving at least one node conforming to Person and (not Barber). If no – if Bob does not conform to Barber – then Bob satisfies the Barber constraint by shaving such a node. Semantics to avoid such paradoxical situations have been proposed based on stratification [Boneva et al., 2017], partial assignments [Corman et al., 2018], and stable models [Gelfond and Lifschitz, 1988].

Although validating schemata and semantic schemata serve different purposes, they can complement each other. In particular, a validating schema can take into consideration a semantic schema, such that, for example, validation is applied on the data graph including inferences. Taking the class hierarchy of Figure 3.1 and the shapes graph of Figure 3.3, for example, we may define the target of the Event shape as the nodes that are of type Event (the class). If we first apply inferencing with respect to the class hierarchy of the semantic schema, the Event shape would now target EID15 and EID16. The presence of a semantic schema may, however, require adapting the validating schema. Taking into account, for example, the aforementioned class hierarchy would require defining a relaxed cardinality on the type property. Open shapes may also be preferred in such cases rather than enumerating constraints on all possible properties that may be inferred on a node.

Two shapes languages have recently emerged for RDF graphs: Shape Expressions (ShEx), published as a W3C Community Group Report [Prud'hommeaux et al., 2014]; and SHACL (Shapes Constraint Language), published as a W3C Recommendation [Knublauch and Kontokostas, 2017]. These languages support the discussed features (and more) and have been adopted for validating graphs in a number of domains relating to healthcare [Thornton et al., 2019], scientific literature [Hammond et al., 2017], spatial data [Car et al., 2019], amongst others. More details about ShEx and SHACL can be found in the book by Labra Gayo et al. [2018]. A recently proposed language that can be used as a common basis for both ShEx and SHACL reveals their similarities and differences [Labra Gayo et al., 2019]. A similar notion of schema has been proposed by Angles [2018] for property graphs.

We formally define shapes following the conventions of Labra Gayo et al. [2019].

Shape
A shape \(\phi\) is defined as:
\(\phi\) ::= \(\top\) true
      \( | \) \(\datatype{N}\) node belongs to the set of nodes \(N\)
\( | \) \(\Psi_{\mathrm{cond}}\) node satisfies the Boolean condition \(\mathrm{cond}\)
\( | \) \(\phi_1 \wedge \phi_2\) conjunction of shape \(\phi_1\) and shape \(\phi_2\)
\( | \) \(\lnot \phi \) negation of shape \(\phi\)
\( | \) \(@s\) reference to shape with label \(s\)
\( | \) \(\qualified{p}{\phi}{min}{max}\)  between \(min\) and \(max\) outward edges (inclusive) with label \(p\)
to nodes satisfying shape \(\phi\)
where \(min \in \mathbb{N}_{(0)}\), \(max \in \mathbb{N}_{(0)} \cup \{ * \}\), with “\(*\)” indicating unbounded.
Shapes schema
A shapes schema is defined as a tuple \(\Sigma = (\Phi,S,\lambda)\) where \(\Phi\) is a set of shapes, \(S\) is a set of shape labels, and \(\lambda : S \rightarrow \Phi\) is a total function from labels to shapes.

The shapes schema from Figure 3.3 can be expressed as:

Event \(\mapsto\) \(\qualifiedL{name}{\datatypeL{string}}{1}{*}\wedge\qualifiedL{start}{\datatypeL{dateTime}}{1}{1}\wedge\qualifiedL{end}{\datatypeL{dateTime}}{1}{1}\)
\(\qquad\wedge\qualifiedL{type}{\top}{1}{*}\wedge\xrightarrow{venue}\)Venue\(\{1,*\}\)
Venue \(\mapsto\) Place\(\:\wedge\qualifiedL{indoor}{\datatypeL{boolean}}{0}{1}\wedge\xrightarrow{city}\)City\(\{0,1\}\)
City \(\mapsto\) Place\(\:\wedge\qualifiedL{population}{(\datatypeL{int}\wedge \Psi_{>5000})}{0}{1}\)
Place \(\mapsto\) \(\qualifiedL{lat}{\datatypeL{float}}{0}{1}\wedge\qualifiedL{long}{\datatypeL{float}}{0}{1}\)
\(\qquad\wedge\xrightarrow{flight}\)Place\(\{0,*\}\wedge\xrightarrow{bus}\)Place\(\{0,*\}\)

For example, Event is a shape label (an element of \(S\)) that maps to a shape (an element of \(\phi\)). This mapping is defined by \(\lambda\).

In a shapes schema, shapes may refer to other shapes, giving rise to a graph that is sometimes known as the shapes graph [Knublauch and Kontokostas, 2017]. Figure 3.3 illustrates a shapes graph of this form.

The semantics of a shape is defined in terms of the evaluation of that shape over each node of a given data graph. The semantics of a shapes schema, in turn, is the result of evaluating each shape of the schema over each node of a given data graph; the result of this evaluation is a shapes map.

Shapes map
Given a directed edge-labelled graph \(G = (V,E,L)\) and a shapes schema \(\Sigma = (\Phi,S,\lambda)\), a shapes map is a (partial) mapping \(\sigma: V \times S \rightarrow \{ 0, 1 \}\).

The shapes map \(\sigma\) is a way of labelling the nodes of \(G\) with the labels of shapes from \(S\). If \(\sigma(v,s) = 1\), then node \(v\) is labelled \(s\) (possibly amongst other labels); otherwise if \(\sigma(v,s) = 0\), then node \(v\) is not labelled \(s\). The precise semantics depends on whether or not \(\sigma\) is a total or partial mapping: whether or not it is defined for every pair in \(V \times S\). Herein we present the semantics for the more straightforward case wherein \(\sigma\) is assumed to be a total shapes map.

Shape evaluation
Given a shapes schema \(\Sigma \coloneqq (\Phi,S,\lambda)\), a directed edge-labelled graph \(G = (V,E,L)\), a node \(v \in V\) and a total shapes map \(\sigma\), the shape evaluation function \(\semantics{\phi}{G}{v}{\sigma} \in \{ 0 , 1 \}\) is defined as follows:
\(\semantics{\top}{G}{v}{\sigma}\) \(=\) \(1\)
\(\semantics{\datatype{N}}{G}{v}{\sigma}\) \(=\) \(1\) iff \(v \in N\)
\(\semantics{\Psi_{\mathrm{cond}}}{G}{v}{\sigma}\) \(=\) \(1\) iff \(\mathrm{cond}(v)\) is true
\(\semantics{\phi_1 \wedge \phi_2}{G}{v}{\sigma}\) \(=\) \(\min\{\semantics{\phi_1}{G}{v}{\sigma}, \semantics{\phi_2}{G}{v}{\sigma}\}\)
\(\semantics{\lnot \phi}{G}{v}{\sigma}\) \(=\) \(1 - \semantics{\phi}{G}{v}{\sigma}\)
\(\semantics{@s}{G}{v}{\sigma}\) \(=\) \(1\) iff \(\sigma(v,s) = 1\)
\(\semantics{\qualified{p}{\phi}{min}{max}}{G}{v}{\sigma}\) \(=\) \(1\) iff \(min \leq \lvert \{ (v,p,u)\in E \mid \semantics{\phi}{G}{u}{\sigma}=1 \} \rvert \leq max\)
If \(\semantics{\phi}{G}{v}{\sigma} = 1\), then \(v\) is said to satisfy \(\phi\) in \(G\) under \(\sigma\).

Typically for the purposes of validating a graph with respect to a shapes schema, a target is defined that requires certain nodes to satisfy certain shapes.

Shapes target
Given a directed edge-labelled graph \(G = (V,E,L)\) and a shapes schema \(\Sigma = (\Phi,S,\lambda)\), a shapes target \(T \subseteq V \times S\) is a set of pairs of nodes and shape labels from \(G\) and \(\Sigma\), respectively.

The nodes that a shape targets can be selected a manual selection, based on the type(s) of the nodes, based on the results of a graph query, etc. [Corman et al., 2018, Labra Gayo et al., 2019].

Lastly, we define the notion of a valid graph under a given shapes schema and target based on the existence of a shapes map satisfying certain conditions.

Valid graph
Given a shapes schema \(\Sigma = (\Phi,S,\lambda)\), a directed edge-labelled graph \(G = (V,E,L)\), and a shapes target \(T\), we say that \(G\) is valid under \(\Sigma\) and \(T\) if and only if there exists a shapes map \(\sigma\) such that, for all \(s \in S\) and \(v \in V\) it holds that \(\sigma(v,s) = \semantics{\lambda(s)}{G}{v}{\sigma}\), and \((v,s) \in T\) implies \(\sigma(v,s) = 1\).

Taking the graph \(G\) from Figure 2.1 and the shapes schema \(\Sigma\) from Figure 3.3, first assume an empty shapes target \(T = \{\}\). If we consider a shapes map where (e.g.) \(\sigma(\)EID15, Event\() = 1\), \(\sigma(\)Santa Lucía, Venue\() = 1\), \(\sigma(\)Santa Lucía, Place\() = 1\), etc., but where \(\sigma(\)EID16, Event\() = 0\) (as it does not have the required values for start and end), etc., then we see that \(G\) is valid under \(\Sigma\) and \(T\). However, if we were to define a shapes target \(T\) to ensure that the Event shape targets EID15 and EID16 – i.e., to define \(T\) such that \(\{ (\)EID15, Event\(), (\)EID16, Event\() \} \subseteq T\) – then the graph would no longer be valid under \(\Sigma\) and \(T\) since EID16 does not satisfy Event.

The semantics we present here assumes that each node in the graph either satisfies or does not satisfy each shape labelled by the schema. More complex semantics – for example, based on Kleene’s three-valued logic [Corman et al., 2018, Labra Gayo et al., 2019] – have been proposed that support partial shapes maps, where the satisfaction of some nodes for some shapes can be left as undefined. Shapes languages in practice may support other more advanced forms of constraints, such as counting on paths [Knublauch and Kontokostas, 2017]. In terms of implementing validation with respect to shapes, work has been done on translating constraints into sets of graph queries, whose results are input to a SAT solver for recursive cases [Corman et al., 2019].

Emergent schema

Both semantic and validating schemata require a domain expert to explicitly specify definitions and constraints. However, a data graph will often exhibit latent structures that can be automatically extracted as an emergent schema [Pham et al., 2015] (aka graph summary [Liu et al., 2018, Čebirić et al., 2019, Spahiu et al., 2016]).

A framework often used for defining emergent schema is that of quotient graphs, which partition groups of nodes in the data graph according to some equivalence relation while preserving some structural properties of the graph. Taking Figure 2.1, we can intuitively distinguish different types of nodes based on their context, such as event nodes, which link to venue nodes, which in turn link to city nodes, and so forth. In order to describe the structure of the graph, we could consider six partitions of nodes: event, name, venue, class, date-time, city. In practice, these partitions may be computed based on the class or shape of the node. Merging the nodes of each partition into one node while preserving edges leads to the quotient graph shown in Figure 3.4: the nodes of this quotient graph are the partitions of nodes from the data graph and an edge \(X\)\(y\)\(Z\) is included the quotient graph if and only if there exists \(x \in X\) and \(z \in Z\) such that \(x\)\(y\)\(z\) is in the original data graph.

Example quotient graph simulating the data graph in Figure 1
Example quotient graph simulating the data graph in Figure 2.1

There are many ways in which quotient graphs may be defined, depending not only on how nodes are partitioned, but also how the edges are defined. Different quotient graphs may provide different guarantees with respect to the structure they preserve. Formally, we can say that every quotient graph simulates its input graph (based on the simulation relation of set membership between data nodes and quotient nodes), meaning that for all \(x \in X\) with \(x\) an input node and \(X\) a quotient node, if \(x\)\(y\)\(z\) is an edge in the data graph, then there must exist an edge \(X\)\(y\)\(Z\) in the quotient graph such that \(z \in Z\); for example, the quotient graph of Figure 3.4 simulates the data graph of Figure 2.1. However, this quotient graph seems to suggest (for instance) that EID16 would have a start and end date in the data graph when this is not the case. A stronger notion of structural preservation is given by bisimilarity, which in this case would further require that if \(X\)\(y\)\(Z\) is an edge in the quotient graph, then for all \(x \in X\), there must exist a \(z \in Z\) such that \(x\)\(y\)\(z\) is in the data graph; this is not satisfied by EID16 in the quotient graph of Figure 3.4, which does not have an outgoing edge labelled start or end in the original data graph. Figure 3.5 illustrates a bisimilar version of the quotient graph, splitting the event partition into two nodes reflecting their different outgoing edges. An interesting property of bisimilarity is that it preserves forward-directed paths: given a path expression \(r\) without inverses and two bisimilar graphs, \(r\) will match a path in one graph if and only if it matches a corresponding path in the other bisimilar graph. One can verify, for example, that a path matches \(x\)city\(\cdot\)(flight|bus)*\(z\) in Figure 2.1 if and only if there is a path matching \(X\)city\(\cdot\)(flight|bus)*\(Z\) in Figure 3.5 such that \(x \in X\) and \(z \in Z\).

Example quotient graph bisimilar with the data graph in Figure 1
Example quotient graph bisimilar with the data graph in Figure 2.1

There are many ways in which quotient graphs may be defined, depending on the equivalence relation that partitions nodes. Furthermore, there are many ways in which other similar or bisimilar graphs can be defined, depending on the (bi)simulation relation that preserves the data graph’s structure [Čebirić et al., 2019]. Such techniques aim to summarise the data graph into a higher-level topology. In order to reduce the memory overhead of the quotient graph, in practice, nodes may rather be labelled with the cardinality of the partition and/or a high-level label (e.g., event, city) for the partition rather than storing the labels of all nodes in the partition.

Various other forms of emergent schema not directly based on a quotient graph framework have also been proposed; examples include emergent schemata based on relational tables [Pham et al., 2015], and baseed on formal concept analysis [González and Hogan, 2018]. Emergent schemata may be used to provide a human-understandable overview of the data graph, to aid with the definition of a semantic or validating schema, to optimise the indexing and querying of the graph, to guide the integration of data graphs, and so forth. We refer to the survey by Čebirić et al. [2019] dedicated to the topic for further details.

Emergent schemata are often based on the notion of a quotient graph.

Quotient graph
Given a directed edge-labelled graph \(G = (V,E,L)\), a graph \(\mathcal{G} = (\mathcal{V},\mathcal{E},L)\) is a quotient graph of \(G\) if and only if:
  • \(\mathcal{V}\) is a partition of \(V\) without the empty set, i.e., \(\mathcal{V} \subseteq (2^V - \emptyset)\), \(V = \bigcup_{U\in \mathcal{V}} U\), and for all \(U\in \mathcal{V}\), \(W\in \mathcal{V}\), it holds that \(U = W\) or \(U \cap W = \emptyset\); and
  • \(\mathcal{E} = \{ (U,l,W) \mid U \in \mathcal{V}, W \in \mathcal{V} \text{ and } \exists u \in U, \exists w \in W : (u,l,w) \in E \} \).

A quotient graph can “merge” multiple nodes into one node, keeping the edges of its constituent nodes. For an input graph \(G = (V,E,L)\), there is an exponential number of possible quotient graphs based on partitions of the input nodes. On one extreme, the input graph is a quotient graph of itself (turning nodes like u into singleton nodes like \(\{\)u\(\}\)). On the other extreme, a single node \(V\), with all input nodes, and loops \((V,l,V)\) for each edge-label \(l\) used in the set of input edges \(E\), is also a quotient graph. Quotient graphs typically fall somewhere in between, where the partition \(\mathcal{V}\) of \(V\) is often defined in terms of an equivalence relation \(\sim\) on the set \(V\) such that \(\mathcal{V} \coloneqq {\sim}/V\); i.e., \(\mathcal{V}\) is defined as the quotient set of \(V\) with respect to \(\sim\); for example, we might define an equivalence relation on nodes such that \(u \sim v\) if and only if they have the same set of defined types, where \({\sim}/V\) is then a partition whose parts contain all nodes with the same types. Another way to induce a quotient graph is to define the partition in a way that preserves some of the topology (i.e., connectivity) of the input graph. One way to formally define this idea is through simulation and bisimulation.

Simulation
Given two directed edge-labelled graph \(G = (V,E,L)\) and \(G' = (V',E',L')\), let \(R \subseteq V \times V'\) be a relation between the nodes of \(G\) and \(G'\), respectively. We call \(R\) a simulation on \(G\) and \(G'\) if, for all \((v,v') \in R\), the following holds:
  • if \((v,p,w) \in E\) then there exists \(w'\) such that \((v',p,w') \in E'\) and \((w,w') \in R\).
If a simulation exists on \(G\) and \(G'\), we say that \(G'\) simulates \(G\), denoted \(G \rightsquigarrow G'\).
Bisimulation
If \(R\) is a simulation on \(G\) and \(G'\), we call it a bisimulation if, for all \((v,v') \in R\), the following condition holds:
  • if \((v'p,w') \in E'\) then there exists \(w\) such that \((v,p,w) \in E\) and \((w,w') \in R\).
If a bisimulation exists on \(G\) and \(G'\), we call them bisimilar, denoted \(G \approx G'\).

Bisimulation (\(\approx\)) is then an equivalence relation on graphs. By defining the (bi)simulation relation \(R\) in terms of set membership \(\in\), every quotient graph simulates its input graph, but does not necessarily bisimulate its input graph. This gives rise to the notion of bisimilar quotient graphs.

Figures 3.4 and 3.5 exemplify quotient graphs for the graph of Figure 2.1. Figure 3.4 simulates but is not bisimilar to the data graph. Figure 3.5 is bisimilar to the data graph. Often the goal will be to compute the most concise quotient graph that satisfies a given condition; for example, the nodes without outgoing edges in Figure 3.5 could be merged while preserving bisimilarity.

Identity

Figure 2.1 uses nodes like Santiago, but to which Santiago does this node refer? Do we refer to Santiago de Chile, Santiago de Cuba, Santiago de Compostela, or do we perhaps refer to the indie rock band Santiago? Based on edges such as Santa LucíacitySantiago, we may deduce that it is one of the three cities mentioned (not the rock band), and based on the fact that the graph describes tourist attractions in Chile, we may further deduce that it refers to Santiago de Chile. Without further details, however, disambiguating nodes of this form may rely on heuristics prone to error in more difficult cases. To help avoid such ambiguity, first we may use globally-unique identifiers to avoid naming clashes when the knowledge graph is extended with external data, and second we may add external identity links to disambiguate a node with respect to an external source.

Persistent identifiers

Assume we wished to compare tourism in Chile and Cuba, and we have acquired an appropriate knowledge graph for Cuba similar to the one we have for Chile. We can merge two graphs by taking their union. However, as shown in Figure 3.6, using an ambiguous node like Santiago may yield a naming clash: the node is referring to two different real-world cities in both graphs, where the merged graph indicates that Santiago is a city in both Chile and Cuba (rather than two distinct cities).5note 5 Such a naming clash is not unique to graphs, but could also occur if merging tables, trees, etc. To avoid such clashes, long-lasting persistent identifiers (PIDs) [Hakala, 2010] can be created in order to uniquely identify an entity; examples of PID schemes include Digital Object Identifiers (DOIs) for papers, ORCID iDs for authors, International Standard Book Numbers (ISBNs) for books, Alpha-2 codes for counties, and more besides.

Result of merging two graphs with ambiguous local identifiers
Result of merging two graphs with ambiguous local identifiers

In the context of the Semantic Web, the RDF data model goes one step further and recommends that global Web identifiers be used for nodes and edge labels. However, rather than adopt the Uniform Resource Locators (URLs) used to identify the location of information resources such as webpages, RDF 1.1 proposes to use Internationalised Resource Identifiers (IRIs) to identify non-information resources such as cities or events.6note 6 Uniform Resource Identifiers (URIs) can be Uniform Resource Locators (URLs), used to locate information resources, and Uniform Resource Names (URNs), used to name resources. Internationalised Resource Identifiers (IRIs) are URIs that allow Unicode (e.g., http://example.com/Ñam). Hence, for example, in the RDF representation of the Wikidata [Vrandečić and Krötzsch, 2014] – a knowledge graph proposed to complement Wikipedia, discussed in more detail in Chapter 10 – while the URL https://www.wikidata.org/wiki/Q2887 refers to a webpage that can be loaded in a browser providing human-readable metadata about Santiago, the IRI http://www.wikidata.org/entity/Q2887 refers to the city itself. Distinguishing the identifiers for the webpage and the city itself avoids naming clashes; for example, if we use the URL to identify both the webpage and the city, we may end up with an edge in our graph, such as (with readable labels below the edge):

https://www.wikidata.org/wiki/Q2887https://www.wikidata.org/wiki/Property:P112https://www.wikidata.org/wiki/Q203534
[Santiago (URL)][founded by (URL)] [Pedro de Valdivia (URL)]

Such an edge leaves ambiguity: was Pedro de Valdivia the founder of the webpage, or the city? Using IRIs for entities distinct from the URLs for the webpages that describe them avoids such ambiguous cases, where Wikidata thus rather defines the previous edge using less ambiguous identifiers, as follows:

https://www.wikidata.org/entity/Q2887https://www.wikidata.org/prop/direct/P112https://www.wikidata.org/entity/Q203534
[Santiago (IRI)][founded by (IRI)] [Pedro de Valdivia (IRI)]

using IRIs for the city, person, and founder of, distinct from the webpages describing them. These Wikidata identifiers use the prefix http://www.wikidata.org/entity/ for entities and the prefix http://www.wikidata.org/prop/direct/ for relations. Such prefixes are known as namespaces, and are often abbreviated with prefix strings, such as wd: or wdt:, where the latter edge can then be written more concisely using such abbreviations as th edge wd:Q2887wdt:P112wd:Q203534.

If HTTP IRIs are used to identify the graph’s entities, when the IRI is looked up (via HTTP), the web-server can return (or redirect to) a description of that entity in formats such as RDF. This further enables RDF graphs to link to related entities described in external RDF graphs over the Web, giving rise to Linked Data [Berners-Lee, 2006, Heath and Bizer, 2011] (discussed in Chapter 9). Though HTTP IRIs offer a flexible and powerful mechanism for issuing global identifiers on the Web, they are not necessarily persistent: websites may go offline, the resources described at a given location may change, etc. In order to enhance the persistence of such identifiers, Persistent URL (PURL) services offer redirects from a central server to a particular location, where the PURL can be redirected to a new location if necessary, changing the address of a document without changing its identifier. The persistence of HTTP IRIs can then be improved by using namespaces defined through PURL services.

External identity links

Assume that the tourist board opts to define the chile: namespace with an IRI such as http://turismo.cl/entity/ on a web-server that they control, allowing nodes such as chile:Santiago – a shortcut for the IRI http://turismo.cl/entity/Santiago – to be looked up over the Web. While using such a naming scheme helps to avoid naming clashes, the use of IRIs does not necessarily help ground the identity of a resource. For example, an external geographic knowledge graph may assign the same city the IRI geo:SantiagoDeChile in their own namespace, where we have no direct way of knowing that the two identifiers refer to the same city. If we merge the two knowledge graphs, we will end up with two distinct nodes for the same city, and thus not integrate their data.

There are a number of ways to ground the identity of an entity. The first is to associate the entity with uniquely-identifying information in the graph, such as its geo-coordinates, its postal code, the year it was founded, etc. Each additional piece of information removes ambiguity regarding which city is being referred to, providing (for example) more options for matching the city with its analogue in external sources. A second option is to use identity links to state that a local entity has the same identity as another coreferent entity found in an external source; an instantiation of this concept can be found in the OWL standard, which defines the owl:sameAs property relating coreferent entities. Using this property, we could state the edge chile:Santiagoowl:sameAsgeo:SantiagoDeChile in our RDF graph, thus establishing an identity link between the corresponding nodes in both graphs. Rather than specifying pairwise identity links between all knowledge graphs, it suffices if two knowledge graphs provide corresponding identity links to the same external knowledge graph, such as DBpedia or Wikidata; for example, if the local knowledge graph provides an identity link to Wikidata indicating chile:Santiagoowl:sameAswd:Q2887, while the remote knowledge graph has the identity link geo:SantiagoDeChileowl:sameAswd:Q2887, then we can infer chile:Santiagoowl:sameAsgeo:SantiagoDeChile. The semantics of owl:sameAs defined by the OWL standard then allows us to combine the data for both nodes. Such semantics will be discussed later in Chapter 4. Ways in which identity links can be computed will also be discussed later in Chapter 8.

Datatypes

Consider the two date-times on the left of Figure 2.1: how should we assign these nodes persistent/global identifiers? Intuitively it would not make much sense, for example, to assign IRIs to these nodes since their syntactic form tells us what they refer to: specific dates and times in March 2020. This syntactic form is further recognisable by machine, meaning that with appropriate software, we could order such values in ascending or descending order, extract the year, etc.

Most practical data models for graphs allow for defining nodes that are datatype values. RDF utilises XML Schema Datatypes (XSD) [Peterson et al., 2012], amongst others, where a datatype node is given as a pair \((l,d)\) where \(l\) is a lexical string, such as "2020-03-29T20:00:00", and \(d\) is an IRI denoting the datatype, such as xsd:dateTime. The node is then denoted "2020-03-29T20:00:00"^^xsd:dateTime. Datatype nodes in RDF are called literals and are not allowed to have outgoing edges. Other datatypes commonly used in RDF data include xsd:string, xsd:integer, xsd:decimal, xsd:boolean, etc. If the datatype is omitted, the value is assumed to be of type xsd:string. Applications built on top of RDF can then recognise these datatypes, parse them into datatype objects, and apply equality checks, normalisation, ordering, transformations, etc., according to their standard definition. In the context of property graphs, Neo4j [Miller, 2013] also defines a set of internal datatypes on property values that includes numbers, strings, Booleans, spatial points, and temporal values.

Lexicalisation

Global identifiers for entities will sometimes have a human-interpretable form, such as chile:Santiago, but the identifier strings themselves do not carry any formal semantic significance. In other cases, the identifiers used may not be human-interpretable by design. In Wikidata, for instance, Santiago de Chile is identified as wd:Q2887, where such a scheme has the advantage of providing better persistence and of not being biased to a particular human language. As a real-world example, the Wikidata identifier for Eswatini (wd:Q1050) was not affected when the country changed its name from Swaziland, and does not necessitate choosing between languages for creating (more readable) IRIs such as wd:Eswatini (English), wd:eSwatini (Swazi), wd:Esuatini (Spanish), etc.

Since identifiers can be arbitrary, it is common to add edges that provide a human-interpretable label for nodes, such as wd:Q2887rdfs:label“Santiago”, indicating how people may refer to the subject node linguistically. Linguistic information of this form plays an important role in grounding knowledge such that users can more clearly identify which real-world entity a particular node in a knowledge graph actually references [de Melo, 2015]; it further permits cross-referencing entity labels with text corpora to find, for example, documents that potentially speak of a given entity [Martínez-Rodríguez et al., 2020]. Labels can be complemented with aliases (e.g., wd:Q2887skos:altLabel“Santiago de Chile”) or comments (e.g. wd:Q2887rdfs:comment“Santiago is the capital of Chile”) to further help ground the node’s identity.

Nodes such as “Santiago” denote string literals, rather than an identifier. Depending on the specific graph model, such literal nodes may also be defined as a pair \((s,l)\), where \(s\) denotes the string and \(l\) a language code; in RDF, for example we may state chile:Cityrdfs:label“City”@en, chile:Cityrdfs:label“Ciudad”@es, etc., indicating labels for the node in different languages. In other models, the pertinent language can rather be specified, e.g., via metadata on the edge. Knowledge graphs with human-interpretable labels, aliases, comments, etc., (in various languages) are sometimes called (multilingual) lexicalised knowledge graphs [Bonatti et al., 2018]".

Existential nodes

When modelling incomplete information, we may in some cases know that there must exist a particular node in the graph with particular relationships to other nodes, but without being able to identify the node in question. For example, we may have two co-located events chile:EID42 and chile:EID43 whose venue has yet to be announced. One option is to simply omit the venue edges, in which case we lose the information that these events have a venue and that both events have the same venue. Another option might be to create a fresh IRI representing the venue, but semantically this becomes indistinguishable from there being a known venue. Hence some graph models permit the use of existential nodes, represented here as a blank circle:

chile:EID42chile:venue     chile:venuechile:EID43

These edges denote that there exists a common venue for chile:EID42 and chile:EID42 without identifying it. Existential nodes are supported in RDF as blank nodes [Cyganiak et al., 2014], which are also commonly used to support modelling complex elements in graphs, such as RDF lists [Cyganiak et al., 2014, Hogan et al., 2014]. Figure 3.7 exemplifies an RDF list, which uses blank nodes in a linked-list structure to encode order. Though existential nodes can be convenient, their presence can complicate operations on graphs, such as deciding if two data graphs have the same structure modulo existential nodes [Cyganiak et al., 2014, Hogan, 2017]. Hence methods for skolemising existential nodes in graphs – replacing them with canonical labels – have been proposed [Longley and Sporny, 2019, Hogan, 2017]. Other authors rather call to minimise the use of such nodes in graph data [Heath and Bizer, 2011].

RDF list representing the three largest peaks of Chile, in order
RDF list representing the three largest peaks of Chile, in order

Context

Many (arguably all) facts presented in the data graph of Figure 2.1 can be considered true with respect to a certain context. With respect to temporal context, Santiago has existed as a city since 1541, flights from Arica to Santiago began in 1956, etc. With respect to geographic context, the graph describes events in Chile. With respect to provenance, data relating to EID15 were taken from – and are thus said to be true with respect to – the Ñam webpage on January 4th, 2020. Other forms of context may also be used. We may further combine contexts, such as to indicate that Arica is a Chilean city (geographic) since 1883 (temporal) per the Treaty of Ancón (provenance).

By context we herein refer to the scope of truth, i.e., the context in which some data are held to be true [McCarthy, 1993, Guha et al., 2004]. The graph of Figure 2.1 leaves much of its context implicit. However, making context explicit can allow for interpreting the data from different perspectives, such as to understand what held true in 2016, what holds true excluding webpages later found to have spurious data, etc. As seen previously, context for graph data may be considered at different levels: on individual nodes, individual edges, or sets of edges (sub-graphs). We now discuss various representations by which context can be made explicit at different levels.

Direct representation

The first way to represent context is to consider it as data no different from other data. For example, the dates for the event EID15 in Figure 2.1 can be seen as representing a form of temporal context, indicating the temporal scope within which edges such as EID15venueSanta Lucía are held true. Another option is to change a relation represented as an edge, such as SantiagoflightArica, into a node, such as seen in Figure 2.3a, allowing us to assign additional context to the relation. While in these examples context is represented in an ad hoc manner, a number of specifications have been proposed to represent context as data in a more standard way. One example is the Time Ontology [Cox et al., 2017], which specifies how temporal entities, intervals, time instants, etc. – and relations between them such as before, overlaps, etc. – can be described in RDF graphs in an interoperable manner. Another example is the PROV Data Model [Gil et al., 2013], which specifies how provenance can be described in RDF graphs, where entities (e.g., graphs, nodes, physical document) are derived from other entities, are generated and/or used by activities (e.g., extraction, authorship), and are attributed to agents (e.g., people, software, organisations).

Reification

Often we may wish to directly define the context of edges themselves; for example, we may wish to state that the edge SantiagoflightArica is valid from 1956. While we could use the pattern of turning the edge into a node – as illustrated in Figure 2.3a – to directly represent such context, another option is to use reification, which allows for making statements about statements in a generic manner (or in the case of a graph, for defining edges about edges). In Figure 3.8 we present three forms of reification that can be used for modelling temporal context on the aforementioned edge within a directed edge-labelled graph [Hernández et al., 2015]. We use \(e\) to denote an arbitrary identifier representing the edge itself to which the context can be associated. Unlike in a direct representation, \(e\) represents an edge, not a flight. RDF reification [Cyganiak et al., 2014] (Figure 3.8a) defines a new node \(e\) to represent the edge and connects it to the source node (via subject), target node (via object), and edge label (via predicate) of the edge. In contrast, \(n\)-ary relations [Cyganiak et al., 2014] (Figure 3.8b) connect the source node of the edge directly to the edge node \(e\) with the label of the edge; the target node of the edge is then connected to \(e\) (via value). Finally, singleton properties [Nguyen et al., 2014] (Figure 3.8c) rather use \(e\) as an edge label, connecting it to a node indicating the original edge label (via singleton). Other forms of reification have been proposed in the literature, including, for example, NdFluents [Giménez-García et al., 2017]. In general, a reified edge does not assert the edge it reifies; for example, we may reify an edge to state that it is no longer valid. We refer to Hernández et al. [2015] for further comparison of reification alternatives.

RDF Reification
RDF Reification
n-ary Relations
\(n\)-ary Relations
Singleton properties
Singleton properties
Three representations of temporal context on a directed labelled edge

Higher-arity representation

As an alternative to reification, we can rather use higher-arity representations for modelling context. Taking again the edge SantiagoflightArica, Figure 3.9 illustrates three higher-arity representations of temporal context. First, we can use a named graph (Figure 3.9a) to contain the edge and then define the temporal context on the graph name. Second, we can use a property graph (Figure 3.9b) where the temporal context is defined as a property on the edge. Third, we can use RDF* [Hartig, 2017] (Figure 3.9c): an extension of RDF that allows edges to be defined as nodes. Amongst these options, the most flexible is the named graph representation, where we can assign context to multiple edges at once by placing them in one named graph; for example, we can add more edges to the named graph of Figure 3.9a that are also valid from 1956. The least flexible option is RDF*, which, in the absence of an edge id, does not permit different groups of contextual values to be assigned to an edge; for example, if we add four contextual values to the edge ChilepresidentM. Bachelet, to state that it was valid from 2006 until 2010 and valid from 2014 until 2018, we cannot pair the values, but may rather have to create a node to represent different presidencies (in the other models, we could have used two named graphs or edge ids).

Named graph
Named graph
Property graph
Property graph
RDF*
RDF*
Three higher-arity representations of temporal context on an edge

Annotations

Thus far, we have discussed representing context in a graph, but we have not spoken about automated mechanisms for reasoning about context; for example, if there are only seasonal summer flights from Santiago to Arica, we may wish to find other routes from Santiago for winter events taking place in Arica. While the dates for buses, flights, etc., can be represented directly in the graph, or using reification, writing a query to manually intersect the corresponding temporal contexts will be difficult. An alternative is to consider annotations that provide mathematical definitions of a contextual domain and key operations over that domain that can be applied automatically.

Some annotations model a particular contextual domain; for example, Temporal RDF [Gutiérrez et al., 2007] allows for annotating edges with time intervals, such as Chilepresident
[2006,2010]
M. Bachelet, while Fuzzy RDF [Straccia, 2009] allows for annotating edges with a degree of truth such as Santiagoclimate
0.8
Semi-Arid, indicating that it is more-or-less true – with a degree of \(0.8\) – that Santiago has a semi-arid climate.

Other forms of annotation are domain-independent; for example, Annotated RDF [Dividino et al., 2009, Udrea et al., 2010, Zimmermann et al., 2012] allows for representing context modelled as semi-rings: algebraic structures consisting of domain values (e.g., temporal intervals, fuzzy values, etc.) and two operators to combine domain values: meet and join.7note 7 The join operator for annotations is different from the join operator for relational algebra. We provide an example in Figure 3.10, where \(G\) is annotated with values from a temporal domain using sets of integers (\(1{-}365\) to represent days of the year. For brevity we use intervals, where, e.g., \(\{[150,152]\}\) denotes the set \(\{150,151,152\}\). Query \(Q\) then asks for flights from Santiago to cities with events; this query will check and return an annotation reflecting the temporal validity of each answer. To derive these answers, we require a conjunction of annotations on compatible flight and city edges, using the meet operator to compute the annotation for which both edges hold. The natural way to define meet here is as the intersection of sets of days, where, for example, applying meet on the event annotation \(\color{blue}\{[150,152]\}\) and the flight annotation \(\color{blue}\{[1,120],[220,365]\}\) for Punta Arenas leads to the empty time interval \(\color{blue}\{\}\), which may thus lead to the city being filtered from the results (depending on the query evaluation semantics). However, for Arica, we find two different non-empty intersections: \(\color{blue}\{[123,125]\}\) for EID16 and \(\color{blue}\{[276,279]\}\) for EID17. Given that we are interested in just the city (a projected variable), we can combine the two annotations for Arica using the join operator, returning the annotation in which either result holds true. The natural way to define join is as the union of the sets of days, giving \(\color{blue}\{[123,125],[276,279]\}\).

Temporally annotated graph Example query
\(Q(G) :\)
?citycontext
Arica\(\color{blue}\{[123,125],[276,279]\}\)
Example query on a temporally annotated graph

We define an annotation domain per Zimmermann et al. [2012].

Annotation domain
Let \(A\) be a set of annotation values. An annotation domain is an idempotent, commutative semi-ring \(D = \langle A,\oplus,\otimes,\bot,\top \rangle\).

This definition can then instantiate specific domains of context.

Letting \(D\) be a semi-ring imposes that, for any values \(a, a_1, a_2, a_3\) in \(A\), the following hold:

  • \((a_1 \oplus a_2) \oplus a_3 = a_1 \oplus (a_2 \oplus a_3)\)
  • \((\bot \oplus a) = (a \oplus \bot) = a\)
  • \((a_1 \oplus a_2) = (a_2 \oplus a_1)\)
  • \((a_1 \oplus a_2) = (a_2 \oplus a_1)\)
  • \((a_1 \otimes a_2) \otimes a_3 = a_1 \otimes (a_2 \otimes a_3)\)
  • \((\top \otimes a) = (a \otimes \top) = a\)
  • \(a_1 \otimes (a_2 \oplus a_3) = (a_1 \otimes a_2) \oplus (a_1 \otimes a_3)\)
  • \((a_1 \oplus a_2) \otimes a_3 = (a_1 \otimes a_3) \oplus (a_2 \otimes a_3)\)
  • \((\bot \otimes a) = (a \otimes \bot) = \bot\)

The requirement that it be idempotent further imposes the following:

  • \((a \oplus a) = a\)

Finally, the requirement that it be commutative imposes the following:

  • \((a_1 \otimes a_2) = (a_2 \otimes a_1)\)

Idempotence induces a partial order: \(a_1 \leq a_2\) if and only if \(a_1 \oplus a_2 = a_2\). Imposing these conditions on the annotation domain allow for reasoning and querying to be conducted over the annotation domain in a well-defined manner. Annotated graphs can then be defined in the natural way:

Annotated directed edge-labelled graph
Letting \(D = \langle A,\oplus,\otimes,\bot,\top \rangle\) denote an idempotent, commutative semi-ring, we define an annotated directed edge-labelled graph (or annotated directed edge-labelled graph) as \(G = (V,E_A,L)\) where \(V \subseteq \con\) is a set of nodes, \(L \subseteq \con\) is a set of edge labels, and \(E_A \subseteq V \times L \times V \times A\) is a set of edges annotated with values from \(A\).

Figure 3.10 exemplifies query answering on a graph annotated with days of the year. Formally this domain can be defined as follows: \(A \coloneqq 2^{\mathbb{N}_{[1,365]}}\), \(\oplus \coloneqq \cup\), \(\otimes \coloneqq \cap\), \(\top \coloneqq \mathbb{N}_{[1,365]}\), \(\bot \coloneqq \emptyset\), where one may verify that \(D = \langle 2^{\mathbb{N}_{[1,365]}}, \cup, \cap, \mathbb{N}_{[1,365]}, \emptyset \rangle\) is indeed an idempotent, commutative semi-ring.

Other contextual frameworks

Other frameworks have been proposed for modelling and reasoning about context in graphs. A notable example is that of contextual knowledge repositories [Serafini and Homola, 2012], which allow for assigning individual (sub-)graphs to their own context. Unlike in the case of named graphs, context is explicitly modelled along one or more dimensions, where each (sub-)graph takes a value for each dimension. Each dimension is associated with a partial order over its values – e.g., 2020-03-22 \(\preceq\) 2020-03 \(\preceq\) 2020 – enabling the selection and combination of sub-graphs that are valid within contexts at different granularities. Schuetz et al. [2021] similarly propose a form of contextual OnLine Analytic Processing (OLAP), based on a data cube formed by dimensions where each cell contains a knowledge graph. Operations such as “slice-and-dice” (selecting knowledge according to given dimensions), as well as “roll-up” (aggregating knowledge at a higher level) are supported. We refer the reader to the respective papers for more details [Serafini and Homola, 2012, Schuetz et al., 2021].

Deductive Knowledge

As humans, we can deduce more from the data graph of Figure 2.1 than what the edges explicitly indicate. We may deduce, for example, that the Ñam festival (EID15) will be located in Santiago, even though the graph does not contain an edge EID15locationSantiago. We may further deduce that the cities connected by flights must have some airport nearby, even though the graph does not contain nodes referring to these airports. In these cases, given the data as premises, and some general rules about the world that we may know a priori, we can use a deductive process to derive new data, allowing us to know more than what is explicitly given by the data. These types of general premises and rules, when shared by many people, form part of “commonsense knowledge” [McCarthy, 1990]; conversely, when rather shared by a few experts in an area, they form part of “domain knowledge”, where, for example, an expert in biology may know that hemocyanin is a protein containing copper that carries oxygen in the blood of some species of Mollusca and Arthropoda.

Machines, in contrast, do not have a priori access to such deductive faculties; rather they need to be given formal instructions, in terms of premises and entailment regimes, facilitating similar deductions to what a human can make. In this way, we will be making more of the meaning (i.e., semantics) of the graph explicit in a machine-readable format. These entailment regimes formalise the conclusions that logically follow as a consequence of a given set of premises. Once instructed in this manner, machines can (often) apply deductions with a precision, efficiency, and scale beyond human performance. These deductions may serve a range of applications, such as improving query answering, (deductive) classification, finding inconsistencies, etc. As a concrete example involving query answering, assume we are interested in knowing the festivals located in Santiago; we may straightforwardly express such a query as per the graph pattern shown in Figure 4.1. This query returns no results for the graph in Figure 2.1: there is no node named Festival, and nothing has (directly) the location Santiago. However, an answer (Ñam) could be automatically entailed were we to state that \(x\) being a Food Festival entails that \(x\) is a Festival, or that \(x\) having venue \(y\) in city \(z\) entails that \(x\) has location \(z\). How, then, should such entailments be captured? In Section 3.1.1 we already discussed how the former entailment can be captured with sub-class relations in a semantic schema; the second entailment, however, requires a more expressive entailment regime than seen thus far.

Graph pattern querying for names of festivals in Santiago
Graph pattern querying for names of festivals in Santiago

In this chapter, we discuss ways in which more complex entailments can be expressed and automated. Though we could leverage a number of logical frameworks for these purposes – such as First-Order Logic, Datalog, Prolog, Answer Set Programming, etc. – we focus on ontologies, which constitute a formal representation of knowledge that, importantly for us, can be represented as a graph. We then discuss how these ontologies can be formally defined, how they relate to existing logical frameworks, and how reasoning can be conducted with respect to such ontologies.

Ontologies

To enable entailment, we must be precise about what the terms we use mean. Returning to Figure 2.1, for example, and examining the node EID16 more closely, we may begin to question how it is modelled, particularly in comparison with EID15. Both nodes – according to the class hierarchy of Figure 3.1 – are considered to be events. But what if, for example, we wish to define two pairs of start and end dates for EID16 corresponding to the different venues? Should we rather consider what takes place in each venue as a different event? What then if an event has various start and end dates in a single venue: would these also be considered as one (recurring) event, or many events? These questions are facets of a more general question: what precisely do we mean by an “event”? Does it happen in one contiguous time interval or can it happen many times? Does it happen in one place or can it happen in multiple? There are no “correct” answers to such questions – we may understand the term “event” in a variety of ways, and thus the answers are a matter of convention.

In the context of computing, an ontology8note 8 The term stems from the philosophical study of ontology, concerning the kinds of entities that exist, the nature of their existence, what kinds of properties they have, and how they may be identified and categorised. is then a concrete, formal representation of what terms mean within the scope in which they are used (e.g., a given domain). For example, one event ontology may formally define that if an entity is an “event”, then it has precisely one venue and precisely one time instant in which it begins. Conversely, a different event ontology may define that an “event” can have multiple venues and multiple start times, etc. Each such ontology formally captures a particular perspective – a particular convention. Under the first ontology, for example, we could not call the Olympics an “event”, while under the second ontology we could. Likewise ontologies can guide how graph data are modelled. Under the first ontology we may split EID16 into two events. Under the second, we may elect to keep EID16 as one event with two venues. Ultimately, given that ontologies are formal representations, they can be used to automate entailment.

Like all conventions, the usefulness of an ontology depends on the level of agreement on what that ontology defines, how detailed it is, and how broadly and consistently it is adopted. Adoption of an ontology by the parties involved in one knowledge graph may lead to a consistent use of terms and consistent modelling in that knowledge graph. Agreement over multiple knowledge graphs will, in turn, enhance the interoperability of those knowledge graphs.

Amongst the most popular ontology languages used in practice are the Web Ontology Language (OWL) [Hitzler et al., 2012]9note 9 We could include RDF Schema (RDFS) in this list, but it is largely subsumed by OWL, which extends its core., recommended by the W3C and compatible with RDF graphs; and the Open Biomedical Ontologies Format (OBOF) [Mungall et al., 2012], used mostly in the biomedical domain. Since OWL is the more widely adopted, we focus on its features, though many similar features are found in both [Mungall et al., 2012]. Before introducing such features, however, we must discuss how graphs are to be interpreted.

Interpretations and models

We as humans may interpret the node Santiago in the data graph of Figure 2.1 as referring to the real-world city that is the capital of Chile. We may further interpret an edge AricaflightSantiago as stating that there are flights from the city of Arica to this city. We thus interpret the data graph as another graph – what we here call the domain graph – composed of real-world entities connected by real-world relations. The process of interpretation, here, involves mapping the nodes and edges in the data graph to nodes and edges of the domain graph.

Along these lines, we can abstractly define an interpretation of a data graph as being composed of two elements: a domain graph, and a mapping from the terms (nodes and edge-labels) of the data graph to those of the domain graph. The domain graph follows the same model as the data graph; for example, if the data graph is a directed edge-labelled graph, then so too will be the domain graph. For simplicity, we will speak of directed edge-labelled graphs and refer to the nodes of the domain graph as entities, and to its edges as relations. Given a data graph and an interpretation, while we denote nodes in the data graph by Santiago, we will denote the entity it refers to in the domain graph by Santiago (per the mapping of the given interpretation). Likewise, while we denote an edge by AricaflightSantiago, we will denote the relation by Aricaflightarrow tip rightwardSantiago (again, per the mapping of the given interpretation). In this abstract notion of an interpretation, we do not require that Santiago or Arica be the real-world cities, nor even that the domain graph contain real-world entities and relations: an interpretation can have any domain graph and mapping.

Why is such an abstract notion of interpretation useful? The distinction between nodes/edges and entities/relations becomes important when we define the meaning of ontology features and entailment. To illustrate this distinction, if we ask whether there is an edge labelled flight between Arica and Viña del Mar for the data graph in Figure 2.1, the answer is no. However, if we ask if the entities Arica and Viña del Mar are connected by the relation flight, then the answer depends on what assumptions we make when interpreting the graph. Under the Closed World Assumption (CWA), if we do not have additional knowledge, then the answer is a definite no – since what is not known is assumed to be false. Conversely, under the Open World Assumption (OWA), we cannot be certain that this relation does not exist as this could be part of some knowledge not (yet) described by the graph. Likewise under the Unique Name Assumption (UNA), the data graph describes at least two flights to Santiago (since Viña del Mar and Arica are assumed to be different entities and, therefore, Aricaflightarrow tip rightwardSantiago and Viña del Marflightarrow tip rightwardSantiago must be different edges). Conversely, under No Unique Name Assumption (NUNA), we can only say that there is at least one such flight since Viña del Mar and Arica may be the same entity with two “names”.

These assumptions (or lack thereof) define which interpretations are valid, and which interpretations satisfy which data graphs. We call an interpretation that satisfies a data graph a model of that data graph. The UNA forbids interpretations that map two data terms to the same domain term. The NUNA allows such interpretations. Under the CWA, an interpretation that contains an edge xparrow tip rightwardy in its domain graph can only satisfy a data graph from which we can entail xpy. Under the OWA, an interpretation containing the edge xparrow tip rightwardy can satisfy a data graph not entailing xpy so long it does not explicitly contradict that edge. OWL adopts the NUNA and OWA, which is the most general case: multiple nodes/edge-labels in the graph may refer to the same entity/relation-type (per the NUNA), and anything not entailed by the data graph is not assumed to be false as a consequence (per the OWA).

A graph interpretation – or simply interpretation – captures the assumptions under which the semantics of a graph can be defined. We define interpretations for directed edge-labelled graphs, though the notion extends naturally to other graph models (assuming the data and domain graphs follow the same model).

Graph interpretation
A (graph) interpretation \(I\) is defined as a pair \(I \coloneqq (\Gamma,\inp{\cdot})\) where \(\Gamma = (V_\Gamma,E_\Gamma,L_\Gamma)\) is a (directed edge-labelled) graph called the domain graph and \(\inp{\cdot} : \con \rightarrow V_\Gamma \cup L_\Gamma\) is a partial mapping from constants to terms in the domain graph.

We denote the domain of the mapping \(\inp{\cdot}\) by \(\textrm{dom}(\inp{\cdot})\). For interpretations under the UNA, the mapping \(\inp{\cdot}\) is required to be injective, while with no UNA (NUNA), no such requirement is necessary.

Interpretations that satisfy a graph are then said to be models of that graph.

Graph models
Let \(G \coloneqq (V,E,L)\) be a directed edge-labelled graph. An interpretation \(I \coloneqq (\Gamma,\inp{\cdot})\) satisfies \(G\) if and only if the following hold:
  • \(V \cup L \subseteq \textrm{dom}(\inp{\cdot})\);
  • for all \(v \in V\), it holds that \(\inp{v} \in V_\Gamma\);
  • for all \(l \in L\), it holds that \(\inp{l} \in L_\Gamma\); and
  • for all \((u,l,v) \in E\), it holds that \((\inp{u},\inp{l},\inp{v}) \in E_\Gamma\).
If \(I\) satisfies \(G\) we call \(I\) a (graph) model of \(G\).

Ontology features

Beyond our base assumptions, we can associate certain patterns in the data graph with semantic conditions that define which interpretations satisfy it; for example, we can add a semantic condition to enforce that if our data graph contains the edge psubp. ofq, then any edge xparrow tip rightwardy in the domain graph of the interpretation must also have a corresponding edge xqarrow tip rightwardy to satisfy the data graph. These semantic conditions then form the features of an ontology language. In what follows, to aid readability, we will introduce the features of OWL using an abstract graphical notation with abbreviated terms. For details of concrete syntaxes, we rather refer to the OWL and OBOF standards [Hitzler et al., 2012, Mungall et al., 2012]. Likewise we present semantic conditions over interpretations for each feature in the same graphical format;10note 10 We abbreviate “if and only if” as “iff” whereby “\(\phi\) iff \(\psi\)” can be read as “if \(\phi\) then \(\psi\)” and “if \(\psi\) then \(\phi\)”. further details of these conditions will be described later in Section 4.1.3.

Individuals

In Table 4.1, we list the main features supported by OWL for describing individuals (e.g., Santiago, EID16), sometimes distinguished from classes and properties. First, we can assert (binary) relations between individuals using edges such as Santa LucíacitySantiago. In the condition column, when we write \(x\)\(y\)arrow tip rightward\(z\), for example, we refer to the condition that the relation is given in the domain graph of the interpretation; if so, the interpretation satisfies the axiom. OWL further allows for defining relations to explicitly state that two terms refer to the same entity, where, e.g., Región Vsame asRegión de Valparaíso states that both refer to the same region (per Section 3.2); or that two terms refer to different entities, where, e.g., Valparaísodiff. fromRegión de Valparaíso distinguishes the city from the region of the same name. We may also state that a relation does not hold using negation, which can be serialised as a graph using a form of reification (see Figure 3.8a).

Ontology features for individuals
Feature Axiom Condition Example
Assertion \(x\)\(y\)\(z\) \(x\)\(y\)arrow tip rightward\(z\) CitycapitalSantiago
Negation negation axiom not \(x\)\(y\)arrow tip rightward\(z\) negation example
Same As \(x_1\)same as\(x_2\) \(x_1\) = \(x_2\) Región Vsame asRegión de Valparaíso
Different From \(x_1\)diff. from\(x_2\) \(x_1\)\(x_2\) Valparaísodiff. fromRegión de Valparaíso
Properties

In Section 3.1.1, we already discussed how sub-properties, domains and ranges may be defined for properties. OWL allows such definitions, and further includes other features, as listed in Table 4.2. We may define a pair of properties to be equivalent, inverses, or disjoint. We can further define a particular property to denote a transitive, symmetric, asymmetric, reflexive, or irreflexive relation. We can also define the multiplicity of the relation denoted by properties, based on being functional (many-to-one) or inverse-functional (one-to-many). We may further define a key for a class, denoting the set of properties whose values uniquely identify the entities of that class. Without adopting a Unique Name Assumption (UNA), from these latter three features we may conclude that two or more terms refer to the same entity. Finally, we can relate a property to a chain (a path expression only allowing concatenation of properties) such that pairs of entities related by the chain are also related by the given property. Note that for the latter two features in Table 4.2 we require representing a list, denoted with a vertical notation ; while such a list may be serialised as a graph in a number of concrete ways, OWL uses RDF lists (see Figure 3.7).

Ontology features for property axioms
Feature Axiom Condition (for all \(x_*\), \(y_*\), \(z_*\)) Example
Sub-property \(p\)subp. of\(q\) \(x\)\(p\)arrow tip rightward\(y\) implies \(x\)\(q\)arrow tip rightward\(y\) venuesubp. oflocation
Domain \(p\)domain\(c\) \(x\)\(p\)arrow tip rightward\(y\) implies \(x\)typearrow tip rightward\(c\) venuedomainEvent
Range \(p\)range\(c\) \(x\)\(p\)arrow tip rightward\(y\) implies \(y\)typearrow tip rightward\(c\) venuerangeVenue
Equivalence \(p\)equiv. p.\(q\) \(x\)\(p\)arrow tip rightward\(y\) iff \(x\)\(q\)arrow tip rightward\(y\) startequiv. p.begins
Inverse \(p\)inv. of\(q\) \(x\)\(p\)arrow tip rightward\(y\) iff \(y\)\(q\)arrow tip rightward\(x\) venueinv. ofhosts
Disjoint \(p\)disj. p.\(q\) not disjoint condition venuedisj. p.hosts
Transitive \(p\)typeTransitive \(x\)\(p\)arrow tip rightward\(y\)\(p\)arrow tip rightward\(z\)
      implies \(x\)\(p\)arrow tip rightward\(z\)
part oftypeTransitive
Symmetric \(p\)typeSymmetric \(x\)\(p\)arrow tip rightward\(y\) iff \(y\)\(p\)arrow tip rightward\(x\) nearbytypeSymmetric
Asymmetric \(p\)typeAsymmetric not asymmetric condition capitaltypeAsymmetric
Reflexive \(p\)typeReflexive reflexive condition part oftypeReflexive
Irreflexive \(p\)typeIrreflexive not irreflexive condition flighttypeIrreflexive
Functional \(p\)typeFunctional \(y_1\)arrow tip leftward\(p\)\(x\)\(p\)arrow tip rightward\(y_2\)
      implies \(y_1\) = \(y_2\)
populationtypeFunctional
Inv. Functional \(p\)typeInv. Functional \(x_1\)\(p\)arrow tip rightward\(y\)arrow tip leftward\(p\)\(x_2\)
      implies \(x_1\) = \(x_2\)
capitaltypeInv. Functional
Key \(c\)key\(p_1\)

\(p_n\)
key condition premise implies \(x_1\)=\(x_2\) Citykeylat
long
Chain \(p\)chain\(q_1\)

\(q_n\)
\(x\)\(q_1\)arrow tip rightward\(y_1\)arrow tip rightward\(y_{n-1}\)\(q_n\)arrow tip rightward\(z\)
      implies \(x\)\(p\)arrow tip rightward\(z\)
locationchainlocation
part of
Classes

In Section 3.1.1, we discussed how class hierarchies can be modelled using a sub-class relation. OWL supports sub-classes, and many additional features, for defining and making claims about classes; these additional features are summarised in Table 4.3. Given a pair of classes, OWL allows for defining that they are equivalent, or disjoint. Thereafter, OWL provides a variety of features for defining novel classes by applying set operators on other classes, or based on conditions that the properties of its instances satisfy. First, using set operators, one can define a novel class as the complement of another class, the union or intersection of a list (of arbitrary length) of other classes, or as an enumeration of all of its instances. Second, by placing restrictions on a particular property \(p\), one can define classes whose instances are all of the entities that have: some value from a given class on \(p\); all values from a given class on \(p\);11note 11 While something like flightpropDomesticAirportallNationalFlight might appear to be a more natural example for All Values, this would be problematic as the corresponding for all condition is satisfied when no such node exists, so we would infer anything known not to have any flights to be a domestic airport. (We could, however, define the intersection of such a definition and airport as being a domestic airport.) have a specific individual as a value on \(p\) (has value); have themselves as a reflexive value on \(p\) (has self); have at least, at most or exactly some number of values on \(p\) (cardinality); and have at least, at most or exactly some number of values on \(p\) from a given class (qualified cardinality). For the latter two cases, in Table 4.3, we use the notation “\(\#\{\)a\(\mid \phi \}\)” to count distinct entities satisfying \(\phi\) in the interpretation. These features can then be combined to create more complex classes, where combining the examples for Intersection and Has Self in Table 4.3 gives the definition: self-driving taxis are taxis having themselves as a driver.

Ontology features for class axioms and definitions
Feature Axiom Condition (for all \(x_*\), \(y_*\), \(z_*\)) Example
Sub-class \(c\)subc. of\(d\) \(x\)typearrow tip rightward\(c\) implies \(x\)typearrow tip rightward\(d\) Citysubc. ofPlace
Equivalence \(c\)equiv. c.\(d\) \(x\)typearrow tip rightward\(c\) iff \(x\)typearrow tip rightward\(d\) Humansuc. ofPerson
Disjoint \(c\)disj. c.\(d\) not \(c\)arrow tip leftwardtype\(x\)typearrow tip rightward\(d\) Citydisj. c.Region
Complement \(c\)comp.\(d\) \(x\)typearrow tip rightward\(c\) iff not \(x\)typearrow tip rightward\(d\) Deadcom.Alive
Union \(c\)union\(d_1\)

\(d_n\)
\(x\)typearrow tip rightward\(c\) iff
\(x\)typearrow tip rightward\(d_1\) or
\(x\)typearrow tip rightward or
\(x\)typearrow tip rightward\(d_n\)
FlightunionDomesticFlight
InternationalFlight
Intersection \(c\)inter.\(d_1\)

\(d_n\)
\(x\)typearrow tip rightward\(c\) iff intersection condition equiv SelfDrivingTaxiinter.Taxi
SelfDriving
Enumeration \(c\)one of\(x_1\)

\(x_n\)
\(x\)typearrow tip rightward\(c\) iff \(x\) \(\in \{\)\(x_1\)\(,\dots,\)\(x_n\)\(\}\) EUStateone ofAustria

Sweden
Some Values some values axiom \(x\)typearrow tip rightward\(c\) iff
there exists \(a\) such that
\(x\)\(p\)arrow tip rightward\(a\)typearrow tip rightward\(d\)
some values example
All Values all values axiom \(x\)typearrow tip rightward\(c\) iff
for all \(a\) with \(x\)\(p\)arrow tip rightward\(a\)
it holds that \(a\)typearrow tip rightward\(d\)
all values example
Has Value has value axiom \(x\)typearrow tip rightward\(c\) iff \(x\)\(p\)arrow tip rightward\(y\) has value example
Has Self has self axiom \(x\)typearrow tip rightward\(c\) iff \(x\)\(p\)arrow tip rightward\(x\) has self example
Cardinality
\(\star \in \{ =, \leq, \geq \}\)
cardinality axiom \(x\)typearrow tip rightward\(c\)
      iff \(\#\{\)a \(\mid\) \(x\)\(p\)arrow tip rightward\(a\)\(\} \star n\)
cardinality example
Qualified
Cardinality
\(\star \in \{ =, \leq, \geq \}\)
qualified cardinality axiom \(x\)typearrow tip rightward\(c\)
      iff \(\#\{\)a \(\mid\) \(x\)\(p\)arrow tip rightward\(a\)typearrow tip rightward\(d\)\(\} \star n\)
qualified cardinality example
Other features

OWL supports other language features not previously discussed, including: annotation properties, which provide metadata about ontologies, such as versioning info; datatype vs. object properties, which distinguish properties that take datatype values from those that do not; and datatype facets, which allow for defining new datatypes by applying restrictions to existing datatypes, such as to define that places in Chile must have a float between \(-66.0\) and \(-110.0\) as their value for the (datatype) property latitude. For more details we refer to the OWL 2 standard [Hitzler et al., 2012]. We will further discuss methodologies for the creation of ontologies in Section 6.5.

Models under semantic conditions

Each axiom described by the previous tables, when added to a graph, enforces some condition(s) on the models the graph. If we were to consider only the base condition of the Assertion feature in Table 4.1, for example, then the models of a graph would be any interpretation such that for every edge xyz in the graph, there exists a relation xyarrow tip rightwardz in the model. Given that there may be other relations in the model (under the OWA), the number of models of any such graph is infinite. Furthermore, given that we can map multiple nodes in the graph to one entity in the model (under the NUNA), any interpretation with (for example) the relation aaarrow tip rightwarda is a model of any graph so long as for every edge xyz in the graph, it holds that x = y = z = a in the interpretation (in other words, the interpretation maps everything to a). As we add axioms with their associated conditions to the graph, we restrict models for the graph; for example, considering a graph with two edges – xyz and ytypeIrreflexive – the interpretation with aaarrow tip rightwarda, x = y = … = a is no longer a model as it breaks the condition for the irreflexive axiom. In this way, we can define a precise model-theoretic semantics for graphs based on how the aforementioned ontological features used in the graph restrict the models of that graph.

We define models under semantics conditions.

Semantic condition
Let \(2^G\) denote the set of all (directed edge-labelled) graphs. A semantic condition is a mapping \(\phi : 2^{G} \rightarrow \{ \text{true}, \text{false} \}\). An interpretation \(I \coloneqq (\Gamma,\inp{\cdot})\) is a model of \(G\) under \(\phi\) if and only if \(I\) is a model of \(G\) and \(\phi(\Gamma)\). Given a set of semantic conditions \(\Phi\), we say that \(I\) is a model of \(G\) if and only if \(I\) is a model of \(G\) and for all \(\phi \in \Phi\), \(\phi(\Gamma)\) is true.

We do not restrict the language used to define semantic conditions, but, for example, we can define the Has Value semantic condition of Table 4.3 in FOL as:

\(\forall c, p, y \Big( \big( \Gamma(c,\)prop\(,p) \wedge \Gamma(c,\)value\(,y) \big) \leftrightarrow \forall x \big( \Gamma(x,\)type\(,c) \leftrightarrow \Gamma(x,p,y) \big) \Big)\)

Here we overload \(\Gamma\) as a ternary predicate to capture the edges of \(\Gamma\). The other semantic conditions enumerated in Tables 4.14.3 can be defined in a similar way [Schneider and Sutcliffe, 2011].12note 12 Although these tables consider axioms originating in the data graph, it suffices to check their image in the domain graph since \(I\) only satisfies \(G\) if the edges of \(G\) defining the axioms are reflected in the domain graph of \(I\) per Definition 4.2. This then simplifies the definitions considerably. This FOL formula defines an if-and-only-if version of the semantic condition for Has Value (described in Section 4.1.4).

Entailment

The conditions listed in the previous tables give rise to entailments, where, for example, in reference to the Symmetric feature of Table 4.2, the definition nearbytypeSymmetric and edge SantiagonearbySantiago Airport entail the edge Santiago AirportnearbySantiago according to the condition given for that feature. We now describe how these conditions lead to entailments.

We say that one graph entails another if and only if any model of the former graph is also a model of the latter graph. Intuitively this means that the latter graph says nothing new over the former graph and thus holds as a logical consequence of the former graph. For example, consider the graph SantiagotypeCitysubc. ofPlace and the graph SantiagotypePlace. All models of the latter must have that Santiagotypearrow tip rightwardPlace, but so must all models of the former, which must have Santiagotypearrow tip rightwardCitysubc. ofarrow tip rightwardPlace and further must satisfy the condition for Sub-class, which requires that Santiagotypearrow tip rightwardPlace also hold. Hence we conclude that any model of the former graph must be a model of the latter graph, or, in other words, the former graph entails the latter graph.

We now formally define entailment under semantic conditions.

Graph entailment
Letting \(G_1\) and \(G_2\) denote two (directed edge-labelled) graphs, and \(\Phi\) a set of semantic conditions, we say that \(G_1\) entails \(G_2\) under \(\Phi\) – denoted \(G_1 \models_\Phi G_2\) – if and only if any model of \(G_1\) under \(\Phi\) is also a model of \(G_2\) under \(\Phi\).

An example of entailment is discussed in Section 4.1.3.13note 13 Here we have defined entailment under OWA. To define entailment under CWA, let \(G \models_\Phi (s,p,o)\) denote that \(G\) entails the edge \((s,p,o)\) under \(\Phi\) (a slight abuse of notation). Under CWA, we make the additional assumption that if \(G \not\models_\Phi e\), where \(e\) is an edge (strictly speaking, a positive edge), then \(G \models_\Phi \neg e\); in other words, under CWA we assume that any (positive) edges that \(G\) does not entail under \(\Phi\) can be assumed false according to \(G\) and \(\Phi\). However, note that in FOL, the CWA only applies to positive facts, whereas edges in a graph can be used to represent other FOL formulae. If one wished to maintain FOL-compatibility under CWA, additional restrictions on the types of edge \(e\) may be needed.

If–then vs. if-and-only-if semantics

Consider the graph nearbytypeSymmetric and the graph nearbyinv. ofnearby. Both of these graphs result in the same semantic conditions being applied in the domain graph, but does one entail the other? The answer depends on the semantics applied. Considering the axioms and conditions of Tables 4.1, we can consider two semantics. Under ifthen semantics – if Axiom matches the data graph then Condition holds in domain graph – the graphs do not entail each other: though both graphs give rise to the same condition, this condition is not translated back into the axioms that describe it.14note 14 Here, nearbytypearrow tip rightwardSymmetric is a model of the first graph but not the second, while nearbyinv. ofarrow tip rightwardnearby is a model of the second graph but not the first. Hence neither graph entails the other. Conversely, under if-and-only-if semantics – Axiom matches data graph if-and-only-if Condition holds in domain graph – the graphs entail each other: both graphs give rise to the same condition, which is translated back into all possible axioms that describe it. Hence if-and-only-if semantics allows for entailing more axioms in the ontology language than if–then semantics. OWL generally applies an if-and-only-if semantics in order to enable richer entailments [Hitzler et al., 2012].

Reasoning

Unfortunately, given two graphs, deciding if the first entails the second – per the notion of entailment we have defined and for all of the ontological features listed in Tables 4.14.3 – is undecidable: no (finite) algorithm for such entailment can exist that halts on all inputs with the correct true/false answer [Hitzler et al., 2010]. However, we can provide practical reasoning algorithms for ontologies that (1) halt on any pair of input ontologies but may miss entailments, returning false instead of true in some cases, (2) always halt with the correct answer but only accept input ontologies with restricted features, or (3) only return correct answers for any pair of input ontologies but may never halt on certain inputs. Though option (3) has been explored using, e.g., theorem provers for First Order Logic (FOL) [Schneider and Sutcliffe, 2011], options (1) and (2) are more commonly pursued using rules and/or Description Logics. Option (1) generally allows for more efficient and scalable reasoning algorithms and is useful where data are incomplete and having some entailments is valuable. Option (2) may be a better choice in domains – such as medical ontologies – where missing entailments may have undesirable outcomes.

Rules

A straightforward way to provide automated access to the knowledge that can be deduced through (ontological or other forms of) entailments is through inference rules (or simply rules) encoding ifthen-style consequences. A rule is composed of a body (if) and a head (then). Both the body and head are given as graph patterns. A rule indicates that if we can replace the variables of the body with terms from the data graph and form a sub-graph of a given data graph, then using the same replacement of variables in the head will yield a valid entailment. The head must typically use a subset of the variables appearing in the body to ensure that the conclusion leaves no variables unreplaced. Rules of this form correspond to (positive) Datalog [Ceri et al., 1989] in Databases, Horn clauses [Lloyd, 1984] in Logic Programming, etc.

Rules can capture entailments under ontological conditions. In Table 4.4, we list some example rules for sub-class, sub-property, domain and range features [Muñoz et al., 2009]; these rules may be considered incomplete, not capturing, for example, that every class is a sub-class of itself, that every property is a sub-property of itself, etc. A more comprehensive set of rules for the OWL features of Tables 4.14.3 have been defined as OWL 2 RL/RDF [Motik et al., 2012]; these rules are likewise incomplete as such rules cannot fully capture negation (e.g., Complement), existentials (e.g., Some Values), universals (e.g., All Values), or counting (e.g., Cardinality and Qualified Cardinality). Other rule languages have, however, been proposed to support additional such features, including existentials (see, e.g., Datalog\(^\pm\) [Bellomarini et al., 2018]), disjunction (see, e.g., Disjunctive Datalog [Rudolph et al., 2008]), etc.

Example rules for sub-class, sub-property, domain, and range features
Feature Body \(\Rightarrow\) Head
Sub-class (I) ?xtype?csubc. of?d \(\Rightarrow\) ?xtype?d
Sub-class (II) ?dsubc. of?dsubc. of?e \(\Rightarrow\) ?dsubc. of?e
Sub-property (I) sub-proprety (I) body \(\Rightarrow\) ?x?q?y
Sub-property (II) ?psubp. of?qsubp. of?r \(\Rightarrow\) ?psubp. of?r
Domain domain body \(\Rightarrow\) ?xtype?c
Range range body \(\Rightarrow\) ?ytype?c

Rules can be leveraged for reasoning in a number of ways. Materialisation refers to the idea of applying rules recursively to a graph, adding the conclusions generated back to the graph until a fixpoint is reached and nothing more can be added. The materialised graph can then be treated as any other graph. Although the efficiency and scalability of materialisation can be enhanced through optimisations like Rete networks [Forgy, 1982], or using distributed frameworks like MapReduce [Urbani et al., 2012], depending on the rules and the data, the materialised graph may become unfeasibly large to manage. Another strategy is to use rules for query rewriting, which given a query, will automatically extend the query in order to find solutions entailed by a set of rules; for example, taking the schema graph in Figure 3.2 and the rules in Table 4.4, the (sub-)pattern ?xtypeEvent in a given input query would be rewritten to the following disjunctive pattern evaluated on the original graph:

?xtypeEvent \(\cup\) ?xtypeType \(\cup\) ?xtypePeriodic Market \(\cup\) ?xvenue?y

Figure 4.2 provides a more complete example of an ontology that is used to rewrite the query of Figure 4.1; if evaluated over the graph of Figure 2.1, Ñam will be returned as a solution. However, not all of the aforementioned features of OWL can be supported in this manner. The OWL 2 QL profile [Motik et al., 2012] is a subset of OWL designed specifically for query rewriting of this form [Artale et al., 2009].

\(O:\)
     ontology
\(Q(O):\)
    \((\)type Festival \(\cup\) type Food Festival \(\cup\) type Drinks Festival\()\)
\(\Join (\)location Santiago \(\cup\) venue city Santiago\()\)
\(\Join \)  name
Query rewriting example for the query \(Q\) of Figure 4.1

While rules can be used to (partially) capture ontological entailments, they can also be defined independently of an ontology language, capturing entailments for a given domain. In fact, some rules – such as the following – cannot be captured by the ontology features previously seen, as they do not support ways to infer relations from cyclical graph patterns (for computability reasons):

dom flight rule premise \(\Rightarrow\) ?xdomestic flight?y

Various languages allow for expressing rules over graphs – independently or alongside of an ontology language – including: Notation3 (N3) [Berners-Lee and Connolly, 2011], Rule Interchange Format (RIF) [Kifer and Boley, 2013], Semantic Web Rule Language (SWRL) [Horrocks et al., 2004], and SPARQL Inferencing Notation (SPIN) [Knublauch et al., 2011], amongst others.

Given a graph pattern \(Q\) – be it a directed edge-labelled graph pattern per Definition 2.5 or a property graph pattern per Definition 2.6 – recall that \(\var(Q)\) denotes the variables appearing in \(Q\). We now define rules for graphs.

Rule
A rule is a pair \(R \coloneqq (B,H)\) such that \(B\) and \(H\) are graph patterns and \(\var(H) \subseteq B\). The graph pattern \(B\) is called the body of the rule while \(H\) is called the head of the rule.

This definition of a rule applies for directed edge-labelled graphs and property graphs by considering the corresponding type of graph pattern. The head is considered to be a conjunction of edges. Given a graph \(G\), a rule is applied by computing the mappings from the body to the graph and then using those mappings to substitute the variables in \(H\). The restriction \(\var(H) \subseteq B\) ensures that the results of this substitution is a graph, with no variables in \(H\) left unsubstituted.

Rule application
Given a rule \(R = (B,H)\) and a graph \(G\), we define the application of \(R\) over \(G\) as the graph \(R(G) \coloneqq \bigcup_{\mu \in B(G)} \mu(H)\).

Given a set of rules \(\mathcal{R} \coloneqq \{ R_1, \ldots, R_n \}\) and a knowledge graph \(G\), towards defining the set of inferences given by the rules over the graph, we denote by \(\mathcal{R}(G) \coloneqq \bigcup_{R \in \mathcal{R}} R(G)\) the union of the application of all rules of \(\mathcal{R}\) over \(G\), and we denote by \(\mathcal{R}^+(G) \coloneqq \mathcal{R}(G) \cup G\) the extension of \(G\) with respect to the application of \(\mathcal{R}\). Finally, we denote by \(\mathcal{R}^k(G)\) (for \(k \in \mathbb{N^+}\)) the recursive application of \(\mathcal{R}^+(G)\), where \(\mathcal{R}^1(G) \coloneqq \mathcal{R}^+(G)\), and \(\mathcal{R}^{i+1}(G) \coloneqq \mathcal{R}^+(\mathcal{R}^{i}(G))\). We are now ready to define the least model, which captures the inferences possible for \(\mathcal{R}\) over \(G\).

Least model
The least model of \(\mathcal{R}\) over \(G\)} is defined as \(\mathcal{R}^*(G) \coloneqq \bigcup_{k\in \mathbb{N}}(R^k(G))\).

At some point \(R^{k'}(G) = R^{k'+1}(G)\): the rule applications reach a fixpoint and we have the least model. Once the least model \(\mathcal{R}^*(G)\) is computed, the entailed data can be treated as any other data.

Rules can support graph entailments of the form \(G_1 \models_\Phi G_2\). We say that a set of rules \(\mathcal{R}\) is correct for \(\Phi\) if, for any graph \(G\), \(G \models_\Phi \mathcal{R}^*(G)\). We say that \(\mathcal{R}\) is complete for \(\Phi\) if, for any graph \(G\), there does not exist a graph \(G' \not\subseteq \mathcal{R}^*(G)\) such that \(G \models_\Phi G'\). Table 4.4 exemplifies a correct but incomplete set of rules for the semantic conditions of the RDFS standard [Brickley and Guha, 2014].

Alternatively, rather than supporting ontology-based graph entailments, rules can be directly specified in a rule language such as Notation3 (N3) [Berners-Lee and Connolly, 2011], Rule Interchange Format (RIF) [Kifer and Boley, 2013], Semantic Web Rule Language (SWRL) [Horrocks et al., 2004], or SPARQL Inferencing Notation (SPIN) [Knublauch et al., 2011]. Languages such as SPIN represent rules as graphs, allowing the rules of a knowledge graph to be embedded in the data graph. Taking advantage of this fact, we can then consider a form of graph entailment \(G_1 \cup \gamma(\mathcal{R}) \models_\Phi G_2\), where by \(\gamma(\mathcal{R})\) we denote the graph representation of rules \(\mathcal{R}\). If the set of rules \(\mathcal{R}\) is correct and complete for \(\Phi\), we may simply write \(G_1 \cup \gamma(\mathcal{R}) \models G_2\), indicating that \(\Phi\) captures the same semantics for \(\gamma(\mathcal{R})\) as applying the rules in \(\mathcal{R}\). Rules thus offer another form of graph entailment.

Description Logics

Description Logics (DLs) were initially introduced as a way to formalise the meaning of frames  and semantic networks . Since semantic networks are an early version of knowledge graphs, and DLs have heavily influenced the Web Ontology Language, DLs thus hold an important place in the logical formalisation of knowledge graphs. DLs form a family of logics rather than a particular logic. Initially, DLs were restricted fragments of FOL that permit decidable reasoning tasks, such as entailment checking [Baader et al., 2017]. Different DLs strike different balances between expressive power and computational complexity of reasoning. DLs were later extended with features beyond FOL that are useful in the context of modelling graph data, such as transitive closure, datatypes, etc.

DLs are based on three types of elements: individuals, such as Santiago; classes (aka concepts) such as City; and properties (aka roles) such as flight. DLs then allow for making claims, known as axioms, about these elements. Assertional axioms can be either unary class relations on individuals, such as City(Santiago), or binary property relations on individuals, such as flight(Santiago,Arica). Such axioms form the Assertional Box (A-Box). DLs further introduce logical symbols to allow for defining class axioms (forming the Terminology Box, or T-Box for short), and property axioms (forming the Role Box, R-Box); for example, the class axiom City \(\sqsubseteq\) Place states that the former class is a sub-class of the latter one, while the property axiom flight \(\sqsubseteq\) connectsTo states that the former property is a sub-property of the latter one. DLs may then introduce a rich set of logical symbols, not only for defining class and property axioms, but also defining new classes based on existing terms; as an example of the latter, we can define a class \(\exists\)nearby.Airport as the class of individuals that have some airport nearby. Noting that the symbol \(\top\) is used in DLs to denote the class of all individuals, we can then add a class axiom \(\exists\)flight.\(\top \sqsubseteq \exists\)nearby.Airport to state that individuals with an outgoing flight must have some airport nearby. Noting that the symbol \(\sqcup\) can be used in DL to define that a class is the union of other classes, we can further define, for example, that Airport \(\sqsubseteq\) DomesticAirport \(\sqcup\) InternationalAirport, i.e., that an airport is either a domestic airport or an international airport (or both).

The similarities between DL features and the OWL features seen previously are not coincidental: the OWL standard was heavily influenced by DLs, where, for example, the OWL 2 DL language is a fragment of OWL restricted so that entailment becomes decidable, where the restrictions are inspired by those defined for DLs. To exemplify a restriction, DomesticAirport \(\sqsubseteq ~=1\) destination \(\circ\) country.\(\top\) defines in DL syntax that domestic airports have flights destined to precisely one country (where p \(\circ\) q denotes a chain of properties). However, counting chains (in this case with \(=1~\texttt{destination} \circ \texttt{country}\)) is often disallowed in DLs to ensure decidability.

Expressive DLs support complex entailments involving existentials, universals, counting, etc. A common strategy for deciding such entailments is to reduce entailment to satisfiability, which decides if an ontology is consistent or not [Horrocks and Patel-Schneider, 2004].15note 15 \(G\) entails \(G'\) if and only if \(G \cup \text{not}(G')\) is not satisfiable, i.e., it has no model. Thereafter methods such as tableau can be used to check satisfiability, cautiously constructing models by completing them along similar lines to the materialisation strategy previously described, but additionally branching models in the case of disjunction, introducing new elements to represent existentials, etc. If any model is successfully “completed”, the process concludes that the original definitions are satisfiable (see, e.g., [Motik et al., 2009]). Due to their prohibitive computational complexity [Motik et al., 2012] – where for example, disjunction may lead to an exponential number of branching possibilities – such reasoning strategies are not typically applied in the case of large-scale data, though they may be useful when modelling complex domains for knowledge graphs.

A DL knowledge base consists of an A-Box, a T-Box, and an R-Box.

DL knowledge base
DL knowledge base \(\mathsf{K}\) is defined as a tuple \((\mathsf{A},\mathsf{T},\mathsf{R})\), where \(\mathsf{A}\) is the A-Box: a set of assertional axioms; \(\mathsf{T}\) is the T-Box: a set of class (aka concept/terminological) axioms; and \(\mathsf{R}\) is the R-Box: a set of relation (aka property/role) axioms.

Table 4.5 provides definitions for all of the constructs typically found in Description Logics. The syntax column denotes how the construct is expressed in DL. The semantics column defines the meaning of axioms using interpretations, which are defined in a slightly different way to those seen previously for graphs.

DL interpretation
A DL interpretation \(I\) is defined as a pair \((\inpdom,\inp{\cdot})\), where \(\inpdom\) is the interpretation domain, and \(\inp{\cdot}\) is the interpretation function. The interpretation domain is a set of individuals. The interpretation function accepts a definition of either an individual \(a\), a class \(C\), or a relation \(R\), mapping them, respectively, to an element of the domain (\(\inp{a} \in \inpdom\)), a subset of the domain (\(\inp{C} \subseteq \inpdom\)), or a set of pairs from the domain (\(\inp{R} \subseteq \inpdom \times \inpdom\)).

An interpretation \(I\) satisfies a knowledge-base \(\mathsf{K}\) if and only if, for all of the syntactic axioms in \(\mathsf{K}\), the corresponding semantic conditions in Table 4.5 hold for \(I\). In this case, we call \(I\) a model of \(\mathsf{K}\).

For \(\mathsf{K} \coloneqq (\mathsf{A},\mathsf{T},\mathsf{R})\), let:

  • \(\mathsf{A} \coloneqq \{ \)City(Arica), City(Santiago), flight(Arica,Santiago)\(\}\);
  • \(\mathsf{T} \coloneqq \{\)City \(\sqsubseteq\) Place, \(\exists\)flight\(.\top \sqsubseteq \exists\)nearby.Airport\(\} \);
  • \(\mathsf{R} \coloneqq \{\)flight \(\sqsubseteq\) connectsTo\(\} \).

For \(I = (\inpdom,\inp{\cdot})\), let:

  • \(\inpdom \coloneqq \{ ⚓,\,🏔,\,✈ \}\);
  • Arica\(I\) \(\coloneqq\,⚓\), Santiago\(I\) \(\coloneqq\,🏔\), AricaAirport\(I\) \(\coloneqq\,✈\);
  • City\(I\) \(\coloneqq \{ ⚓,\,🏔 \}\), Airport\(I\) \(\coloneqq \{ ✈ \}\);
  • flight\(I\) \(\coloneqq \{ (⚓,\,🏔) \}\), connectsTo\(I\) \(\coloneqq \{ (⚓,\,🏔) \}\), sells\(I\) \(\coloneqq \{ (✈,\,☕) \}\).

The interpretation \(I\) is not a model of \(\mathsf{K}\) since it does not have that \(⚓\) is nearby some Airport, nor that \(⚓\) and \(🏔\) are in the class Place. However, if we extend the interpretation \(I\) with the following:

  • Place\(I\) \(\coloneqq \{ ⚓,\,🏔 \}\);
  • nearby \(\coloneqq \{ (⚓,\,✈) \}\).

Now \(I\) is a model of \(\mathsf{K}\). Note that although \(\mathsf{K}\) does not imply that sells(AricaAirport,coffee) while \(I\) indicates that \(✈\) does indeed sell \(☕\), \(I\) is still a model of \(\mathsf{K}\) since \(\mathsf{K}\) is not assumed to be a complete description, per the OWA.

Finally, the notion of a model gives rise to the notion of entailment, which tells us which knowledge bases hold as a logical consequence of which others.

Entailment
Given two DL knowledge bases \(\mathsf{K}_1\) and \(\mathsf{K}_2\), we define that \(\mathsf{K}_1\) entails \(\mathsf{K}_2\), denoted \(\mathsf{K}_1 \models \mathsf{K}_2\), if and only if any model of \(\mathsf{K}_1\) is a model of \(\mathsf{K}_2\).

Let \(\mathsf{K}_1\) denote the knowledge base \(\mathsf{K}\) from the Example 4.1, and define a second knowledge base with one assertion: \(\mathsf{K}_2 \coloneqq ( \{ \)connectsTo\((\)Arica, Santiago\() \}, \{\}, \{\} )\) with one assertion. Though \(\mathsf{K}_1\) does not assert this axiom, it does entail \(\mathsf{K}_2\): to be a model of \(\mathsf{K}_2\), an interpretation must have that \((\)Arica\(I\), Santiago\() \in\) connectsTo\(I\), but this must also be the case for any interpretation that satisfies \(\mathsf{K}_1\) since it must have that \((\)Arica\(I\), Santiago\(I\)\() \in \)flight and flight \(\subseteq\) connectsTo\(I\). Hence any model of \(\mathsf{K}_1\) must also be a model of \(\mathsf{K}_2\), and \(\mathsf{K}_1 \models \mathsf{K}_2\) holds.

Unfortunately, the problem of deciding entailment for knowledge bases expressed in the DL composed of the unrestricted use of all of the axioms of Table 4.5 is undecidable since we could reduce instances of the Halting Problem to such entailment. Hence DLs in practice restrict use of the features listed in Table 4.5. Different DLs apply different restrictions, implying different trade-offs for expressivity and the complexity of entailment. Most DLs are founded on one of the following base DLs (we use indentation to denote derivation):

  • [\(\mathcal{ALC}\)] (\(\mathcal{A}\)ttributive \(\mathcal{L}\)anguage with \(\mathcal{C}\)omplement} [Schmidt-Schauß and Smolka, 1991]), supports atomic classes, the top and bottom classes, class intersection, class union, class negation, universal restrictions and existential restrictions. Relation and class assertions are also supported.
    • [\(\mathcal{S}\)] extends \(\mathcal{ALC}\) with transitive closure.

These base languages can be extended as follows:

  • [\(\mathcal{H}\)] adds relation inclusion.
    • [\(\mathcal{R}\)] adds (limited) complex relation inclusion, relation reflexivity, relation irreflexivity, relation disjointness and the universal relation.
  • [\(\mathcal{O}\)] adds (limited) nomimals.
  • [\(\mathcal{I}\)] adds inverse relations.
  • [\(\mathcal{F}\)] adds (limited) functional properties.
    • [\(\mathcal{N}\)] adds (limited) number restrictions (covering \(\mathcal{F}\) with \(\top\)).
      • [\(\mathcal{Q}\)] adds (limited) qualified number restrictions (covering \(\mathcal{N}\) with \(\top\)).

We use “(limited)” to indicate that such features are often only allowed under certain restrictions to ensure decidability; for example, complex relations (chains) typically cannot be combined with cardinality restrictions. DLs are then typically named per the following scheme, where \([a|b]\) denotes an alternative between \(a\) and \(b\) and \([c][d]\) denotes a concatenation \(cd\):

\[ [\mathcal{ALC}|\mathcal{S}][\mathcal{H}|\mathcal{R}][\mathcal{O}][\mathcal{I}][\mathcal{F}|\mathcal{N}|\mathcal{Q}] \]

Examples include \(\mathcal{ALCO}\), \(\mathcal{ALCHI}\), \(\mathcal{SHIF}\), \(\mathcal{SROIQ}\), etc. These languages often apply additional restrictions on class and property axioms to ensure decidability, which we do not discuss here. For further details on DLs, we refer to the recent book by Baader et al. [2017].

As mentioned in the body of the survey, DLs have been very influential in the definition of OWL, where the OWL 2 DL fragment (roughly) corresponds to the DL \(\mathcal{SROIQ}\). For example, the axiom venuedomainEvent in OWL can be translated to \(\exists\)venue\(.\top \sqsubseteq\) Event, meaning that the class of individuals with some value for venue (in any class) is a sub-class of the class Event. We leave other translations from the OWL axioms of Tables 4.14.3 to DL as an exercise.16note 16 Though not previously mentioned, OWL additionally defines the classes Thing and Nothing that correspond to \(\top\) and \(\bot\), respectively. Note, however, that axioms like sub-taxon ofsubp. ofsubc. of – which given a graph such as FredtypeHomo sapienssub-taxon ofHominini entails the edge FredtypeHominini – cannot be expressed in DL: “subTaxonOf \(\sqsubseteq\ \sqsubseteq\)” is not syntactically valid. Hence only a subset of graphs can be translated into well-formed DL ontologies; we refer to the OWL standard for details [Hitzler et al., 2012].

Description Logic semantics (such that \(x, y, z, \inp{a}, \inp{a_1}, \ldots \inp{a_n}, \inp{b}\) are in \(\inpdom\))
Name Syntax Semantics (\(\inp{\cdot}\))
Class Definitions
Atomic Class \(A\) \(\inp{A}\) (a subset of \(\inpdom)\)
Top Class \(\top\) \(\inpdom\)
Bottom Class \(\bot\) \(\emptyset\)
Class Negation \(\neg C\) \(\inpdom \setminus \inp{C}\)
Class Intersection \(C \sqcap D\) \(\inp{C} \cap \inp{D}\)
Class Union \(C \sqcup D\) \(\inp{C} \cup \inp{D}\)
Nominal \(\{ a_1, ..., a_n \}\) \(\{ \inp{a_1}, ..., \inp{a_n} \}\)
Existential Restriction \(\exists R.C\) \(\{ x \mid \exists y : (x,y) \in \inp{R}\text{ and }y \in \inp{C} \}\)
Universal Restriction \(\forall R.C\) \(\{ x \mid \forall y : (x,y) \in \inp{R}\text{ implies }y \in \inp{C} \}\)
Self Restriction \(\exists R.\textsf{Self}\) \(\{ x \mid (x,x) \in \inp{R} \}\)
Number Restriction \(\star\,n\,R\) (where \(\star \in \{\geq, \leq, = \}\)) \(\{ x \mid \#\{ y : (x,y) \in \inp{R} \} \star n \}\)
Qualified Number Restriction \(\star\,n\,R.C\) (where \(\star \in \{\geq, \leq, = \}\)) \(\{ x \mid \#\{ y : (x,y) \in \inp{R}\text{ and }y \in \inp{C} \} \star n \}\)
Class Axioms (T-Box)
Class Inclusion \(C \sqsubseteq D\) \(\inp{C} \subseteq \inp{D}\)
Relation Definitions
Relation \(R\) \(\inp{R}\) (a subset of \(\inpdom \times \inpdom\))
Inverse Relation \(R^{-}\) \(\{ (y,x) \mid (x,y) \in \inp{R} \}\)
Universal Relation \(\textsf{U}\) \(\inpdom \times \inpdom\)
Relation Axioms (R-Box)
Relation Inclusion \(R \sqsubseteq S\) \(\inp{R} \subseteq \inp{S}\)
Complex Relation Inclusion \(R_1 \circ ... \circ R_n \sqsubseteq S\) \(\inp{R_1} \circ ... \circ \inp{R_n} \subseteq \inp{S}\)
Transitive Relations \(\textsf{Trans}(R)\) \(\inp{R} \circ \inp{R} \subseteq \inp{R}\)
Functional Relations \(\textsf{Func}(R)\) \(\{ (x,y), (x,z) \} \subseteq \inp{R} \)implies \(y = z\)
Reflexive Relations \(\textsf{Ref}(R)\) for all \(x : (x,x) \in \inp{R}\)
Irreflexive Relations \(\textsf{Irref}(R)\) for all \(x : (x,x) \not\in \inp{R}\)
Symmetric Relations \(\textsf{Sym}(R)\) \(\inp{R} = \inp{(R^{-})}\)
Asymmetric Relations \(\textsf{Asym}(R)\) \(\inp{R} \cap \inp{(R^{-})} = \emptyset\)
Disjoint Relations \(\textsf{Disj}(R,S)\) \(\inp{R} \cap \inp{S} = \emptyset\)
Assertional Definitions
Individual \(a\) \(\inp{a}\)
Assertional Axioms (A-Box)
Relation Assertion \(R(a,b)\) \((\inp{a},\inp{b}) \in \inp{R}\)
Negative Relation Assertion \(\neg R(a,b)\) \((\inp{a},\inp{b}) \not\in \inp{R}\)
Class Assertion \(C(a)\) \(\inp{a} \in \inp{C}\)
Equality \( a = b \) \(\inp{a} = \inp{b}\)
Inequality \( a \neq b \) \(\inp{a} \neq \inp{b}\)

Inductive Knowledge

While deductive knowledge is characterised by precise logical consequences, inductively acquiring knowledge involves generalising patterns from a given set of input observations, which can then be used to generate novel but potentially imprecise predictions. For example, from a large data graph with geographical and flight information, we may observe the pattern that almost all capital cities of countries have international airports serving them, and hence predict that if Santiago is a capital city, it likely has an international airport serving it; however, the predictions drawn from this pattern do not hold for certain, where (e.g.) Vaduz, the capital city of Liechtenstein, has no (international) airport serving it. Hence predictions will often be associated with a level of confidence; for example, we may say that a capital has an international airport in \(\frac{187}{195}\) of cases, offering a confidence of \(0.959\) for predictions made with that pattern. We then refer to knowledge acquired inductively as inductive knowledge, which includes both the models used to encode patterns, as well as the predictions made by those models. Though fallible, inductive knowledge can be highly valuable.

Conceptual overview of popular inductive techniques for knowledge graphs in terms of type of representation generated (Numeric/Symbolic) and type of paradigm used (Unsupervised/Self-supervised/Supervised)
Conceptual overview of popular inductive techniques for knowledge graphs in terms of type of representation generated (Numeric/Symbolic) and type of paradigm used (Unsupervised/Self-supervised/Supervised)

In Figure 5.1 we provide an overview of the inductive techniques typically applied to knowledge graphs. In the case of unsupervised methods, there is a rich body of work on graph analytics, which uses well-known functions/algorithms to detect communities or clusters, find central nodes and edges, etc., in a graph. Alternatively, knowledge graph embeddings can use self-supervision to learn a low-dimensional numeric model of a knowledge graph that (typically) maps input edges to an output plausibility score indicating the likelihood of the edge being true. The structure of graphs can also be directly leveraged for supervised learning, as explored in the context of graph neural networks. Finally, while the aforementioned techniques learn numerical models, symbolic learning can learn symbolic models – i.e., logical formulae in the form of rules or axioms – from a graph in a self-supervised manner. We now discuss each of the aforementioned techniques in turn.

Graph Analytics

Analytics is the process of discovering, interpreting, and communicating meaningful patterns inherent to (typically large) data collections. Graph analytics is then the application of analytical processes to (typically large) graph data. The nature of graphs naturally lends itself to certain types of analytics that derive conclusions about nodes and edges based on the topology of the graph, i.e., how the nodes of the graph are connected. Graph analytics draws upon techniques from related areas, such as graph theory and network analysis, which have been used to study graphs representing social networks, the Web, internet routing, transport networks, ecosystems, protein–protein interactions, linguistic cooccurrences, and more besides [Estrada, 2011].

Returning to the domain of our running example, the tourism board could use graph analytics to extract knowledge about, for instance: key transport hubs that serve many tourist attractions (centrality); groupings of attractions visited by the same tourists (community detection); attractions that may become unreachable in the event of strikes or other route failures (connectivity), or pairs of attractions that are similar to each other (node similarity). Given that such analytics will require a complex, large-scale graph, for the purposes of illustration, in Figure 5.2 we present a more concise example of some transportation connections in Chile directed towards popular tourist destinations. We first introduce a selection of key techniques that can be applied for graph analytics. We then discuss frameworks and languages that can be used to compute such analytics in practice. Given that many traditional graph algorithms are defined for unlabelled graphs, we then describe ways in which analytics can be applied over directed edge-labelled graphs. Finally we discuss the potential connections between graph analytics and querying and reasoning.

Data graph representing transport routes in Chile
Data graph representing transport routes in Chile

Techniques

A wide variety of techniques can be applied for graph analytics. In the following we will enumerate some of the main techniques – as recognised, for example, by the survey of Iosup et al. [2016] – that can be invoked in this setting.

While the previous techniques accept a graph alone as input,17note 17 Node similarity can be run over an entire graph to find the \(k\) most similar nodes for each node, or can also be run for a specific node to find its most similar nodes. There are also measures for graph similarity (based on, e.g., frequent itemsets [Maillot and Bobed, 2018]) that accept multiple graphs as input. other forms of graph analytics may further accept a node, a pair of nodes, etc., along with the graph.

Most of the aforementioned techniques for graph analytics were originally proposed and studied for simple graphs or directed graphs without edge labels. We will discuss their application to more complex graph models – and how they can be combined with other techniques such as reasoning and querying – later in Section 5.1.3.

Frameworks

Various frameworks have been proposed for large-scale graph analytics, often in a distributed (cluster) setting. Amongst these we can mention Apache Spark (GraphX) [Xin et al., 2013a, Dave et al., 2016], GraphLab [Low et al., 2012], Pregel [Malewicz et al., 2010], Signal–Collect [Stutz et al., 2016], Shark [Xin et al., 2013b], etc. These graph parallel frameworks apply a systolic abstraction [Kung, 1982] based on a directed graph, where nodes are seen as processors that can send messages to other nodes along edges. Computation is then iterative, where in each iteration, each node reads messages received through inward edges (and possibly its own previous state), performs a computation, and then sends messages through outward edges based on the result. These frameworks then define the systolic computational abstraction on top of the data graph being processed: nodes and edges in the data graph become nodes and edges in the systolic graph.

To take an example, assume we wish to compute the places that are most (or least) easily reached by the routes shown in the graph of Figure 5.2. A good way to measure this is using centrality, where we choose PageRank [Page et al., 1999], which computes the probability of a tourist randomly following the routes shown in the graph being at a particular place after a given number of “hops”. We can implement PageRank on large graphs using a graph parallel framework. In Figure 5.3, we provide an example of an iteration of PageRank for an illustrative sub-graph of Figure 5.2. The nodes are initialised with a score of \(\frac{1}{|V|} = \frac{1}{6}\), where we assume the tourist to have an equal chance of starting at any point. In the message phase (Msg), each node \(v\) passes a score of \(\frac{d \textrm{R}_i(v)}{|E(v)|}\) on each of its outgoing edges, where we denote by \(d\) a constant damping factor used to ensure convergence (typically \(d = 0.85\), indicating the probability that a tourist randomly “jumps” to any place), by \(\textrm{R}_i(v)\) the score of node \(v\) in iteration \(i\) (the probability of the tourist being at node \(v\) after \(i\) hops), and by \(|E(v)|\) the number of outgoing edges of \(v\). The aggregation phase (Agg) for \(v\) then sums all incoming messages received along with its constant share of the damping factor (\(\frac{1-d}{|V|}\)) to compute \(\textrm{R}_{i+1}(v)\). We then proceed to the message phase of the next iteration, continuing until some termination criterion is reached (e.g., iteration count or residual threshold, etc.) and final scores are output.

Example of a systolic iteration of PageRank for a sample sub-graph of Figure 24
Example of a systolic iteration of PageRank on a sub-graph of Figure 5.2

While the given example is for PageRank, the systolic abstraction is general enough to support a wide variety of graph analytics, including those previously mentioned. An algorithm in this framework consists of the functions to compute message values in the message phase (Msg), and to accumulate the messages in the aggregation phase (Agg). The framework will take care of distribution, message passing, fault tolerance, etc. However, such frameworks – based on message passing between neighbours – have limitations: not all types of analytics can be expressed in such frameworks [Xu et al., 2019].18note 18 Formally, Xu et al. [2019] have shown that such frameworks are as powerful as the (incomplete) Weisfeiler–Lehman (WL) graph isomorphism test for distinguishing graphs. This test involves nodes recursively hashing together hashes of local information received from neighbours, and passing these hashes to neighbours. Hence frameworks may allow additional features, such as a global step that performs a global computation on all nodes, making the result available to each node [Malewicz et al., 2010]; or a mutation step that allows for adding or removing nodes and edges during processing [Malewicz et al., 2010].

Before defining a graph parallel framework, in the interest of generality, we first define a directed graph labelled with feature vectors, which captures the type of input that such a framework can accept, with vectors on both nodes and edges.

Directed vector-labelled graph
We define a directed vector-labelled graph \(G = (V,E,F,\lambda)\), where \(V\) is a set of nodes, \(E \subseteq V \times V\) is a set of edges, \(F\) is a set of feature vectors, and \(\lambda : V \cup E \rightarrow F\) labels each node and edge with a feature vector.

A directed-edge labelled graph or property graph may be encoded as a directed vector-labelled graph in a number of ways. The type of node and/or a selection of its attributes may be encoded in the node feature vectors, while the label of an edge and/or a selection of its attributes may be encoded in the edge feature vector (including, for example, weights applied to edges). Typically node feature vectors will all have the same dimensionality, as will edge feature vectors.

We define a directed vector-labelled graph in preparation for later computing PageRank using a graph parallel framework. Let \(G = (V,E,L)\) denote a directed edge-labelled graph. Let \(|E(u)|\) denote the outdegree of node \(u \in V\). We then initialise a directed vector-labelled graph \(G' = (V,E',F,\lambda)\) such that \(E' = \{ (x,z) \mid \exists y : (x,y,z)\in E \}\), and for all \(u \in V\), we define \(\lambda(u) \coloneqq \begin{bmatrix} \frac{1}{|V|} \\ |E'(u)| \\ |V| \end{bmatrix}\), and \(\lambda(u,v) \coloneqq \begin{bmatrix} \, \end{bmatrix}\), with \(F \coloneqq \{ \lambda(u) \mid u \in V \} \cup \{\lambda(u,v) \mid (u,v) \in E' \}\), assigning each node a vector containing its initial PageRank score, the outdegree of the node, and the number of nodes in the graph. Conversely, edge-vectors are not used in this case.

We now define a graph parallel framework, where we use \(\{\!\!\{ \cdot \}\!\!\}\) to denote a multiset, \(2^{S \rightarrow \mathbb{N}}\) to denote the set of all multisets containing (only) elements from the set \(S\), and \(\mathbb{R}^a\) to denote the set of all vectors of dimension \(a\) (i.e., the set of all vectors containing \(a\) real-valued elements).

Graph parallel framework
A graph parallel framework (GPF) is a triple of functions \(\mathfrak{G} \coloneqq (\)Msg, Agg, End\()\) such that (with \(a, b, c \in \mathbb{N}\)):
  • Msg\(: \mathbb{R}^a \times \mathbb{R}^b \rightarrow \mathbb{R}^c\)
  • Agg\(: \mathbb{R}^a \times 2^{\mathbb{R}^c \rightarrow \mathbb{N}} \rightarrow \mathbb{R}^a\)
  • End\(: 2^{\mathbb{R}^a \rightarrow \mathbb{N}} \rightarrow \{ \mathrm{true}, \mathrm{false} \}\)

The function Msg defines what message (i.e., vector) must be passed from a node to a neighbouring node along a particular edge, given the current feature vectors of the node and the edge; the function Agg is used to compute a new feature vector for a node, given its previous feature vector and incoming messages; the function End defines a condition for termination of vector computation. The integers \(a\), \(b\) and \(c\) denote the dimensions of node feature vectors, edge feature vectors, and message vectors, respectively; we assume that \(a\) and \(b\) correspond with the dimensions of input feature vectors for nodes and edges. Given a GPF \(\mathfrak{G} = (\)Msg, Agg, End\()\), a directed vector-labelled graph \(G = (V, E, F, \lambda)\), and a node \(u \in V\), we define the output vector assigned to node \(u\) in \(G\) by \(\mathfrak{G}\) (written \(\mathfrak{G}(G, u)\)) as follows. First let \(\mathbf{n}_u^{(0)} \coloneqq \lambda(u)\). For all \(i\geq 1\), let:

\begin{align*} M_u^{(i)} & \coloneqq \left\{\!\!\!\left\{ {\rm\small M{\scriptsize SG}}\left(\mathbf{n}_v^{(i-1)},\lambda(v,u)\right) \bigl\lvert\, (v,u) \in E \right\}\!\!\!\right\} \\ \mathbf{n}_{u}^{(i)} & \coloneqq {\rm\small A{\scriptsize GG}}\left(\mathbf{n}_u^{(i-1)},M_u^{(i)}\right) \end{align*}

where \(M_u^{(i)}\) is the multiset of messages received by node \(u\) during iteration \(i\), and \(\mathbf{n}_{u}^{(i)}\) is the state (vector) of node \(u\) at the end of iteration \(i\). If \(j\) is the smallest integer for which End\((\{\!\!\{ \mathbf{n}_u^{(j)} \mid u \in V \}\!\!\})\) is true, then \(\mathfrak{G}(G, u) \coloneqq \mathbf{n}_u^{(j)}\).

This particular definition assumes that vectors are dynamically computed for nodes, and that messages are passed only to outgoing neighbours, but the definitions can be readily adapted to consider dynamic vectors for edges, or messages being passed to incoming neighbours, etc. We now provide an example instantiating a GPF to compute PageRank over a directed graph.

We take as input the directed vector labelled graph \(G' = (V,E,F,\lambda)\) from Example 5.1 for a PageRank GPF. First we define the messages passed from \(u\) to \(v\):

Msg\(\left(\mathbf{n}_v,\lambda(v,u)\right) \coloneqq \begin{bmatrix} \frac{d(\mathbf{n}_{v})_1}{(\mathbf{n}_{v})_2}\\ \end{bmatrix}\)

where \(d\) denotes PageRank’s constant dampening factor (typically \(d \coloneqq 0.85\)) and \((\mathbf{n}_{v})_k\) denotes the \(k\)th element of the \(\mathbf{n}_{v}\) vector. In other words, \(v\) will pass to \(u\) its PageRank score multiplied by the dampening factor and divided by its out-degree (we do not require \(\lambda(v,u)\) in this particular example). Next we define the function for \(u\) to aggregate the messages it receives from other nodes:

Agg\(\left(\mathbf{n}_u,M_u\right) \coloneqq \begin{bmatrix} \frac{1 - d}{(\mathbf{n}_{u})_3} + \sum_{\mathbf{m} \in M_u}(\mathbf{m})_1 \\ (\mathbf{n}_{u})_2 \\ (\mathbf{n}_{u})_3 \\ \end{bmatrix}\)

Here, we sum the scores received from other nodes along with its share of rank from the dampening factor, copying over the node’s degree and the total number of nodes for future use. Finally, there are a number of ways that we could define the termination condition; here we simply define:

End\((\{\!\!\{ \mathbf{n}_u^{(i)} \mid u \in V \}\!\!\}) \coloneqq (i \geq \textsf{z}) \)

where \(\textsf{z}\) is a fixed number of iterations, at which point the process stops.

We may note in this example that the total number of nodes is duplicated in the vector for each node of the graph. Part of the benefit of GPFs is that only local information in the neighbourhood of the node is required for each computation step. In practice, such frameworks may allow additional features, such as global computation steps whose results are made available to all nodes [Malewicz et al., 2010], operations that dynamically modify the graph [Malewicz et al., 2010], etc.

Analytics on data graphs

As aforementioned, most analytics presented thus far are, in their “native” form, applicable for undirected or directed graphs without the edge metadata – i.e., edge labels or property–value pairs – typical of graph data models.19note 19 We remark that in the case of property graphs, property–value pairs on nodes can be converted by mapping values to nodes and properties to edges with the corresponding label. A number of strategies can be applied to make data graphs subject to analytics of this form:

Original graph
Original graph
Lossy transformation
Lossy transformation
Lossless transformation
Lossless transformation
Transformations from a directed edge-labelled graph to a directed graph

The results of an analytical process may change drastically depending on which of the previous strategies are chosen to prepare the graph for analysis. The choice of strategy may be a non-trivial one to make a priori and may require empirical validation. More study is required to more generally understand the effects of such strategies on the results of different analytical techniques over different graph models.

Analytics with queries

As discussed in Section 2.2, various languages for querying graphs have been proposed down through the years [Angles et al., 2017]. One may consider a variety of ways in which query languages and analytics can complement each other. First, we may consider using query languages to project or transform a graph suitable for a particular analytical task, such as to extract the graph of Figure 5.2 from a larger data graph. Query languages such as SPARQL [Harris et al., 2013], Cypher [Francis et al., 2018], and G-CORE [Angles et al., 2018] allow for outputting graphs, where such queries can be used to select sub-graphs for analysis. These languages can also express some limited (non-recursive) analytics, where aggregations can be used to compute degree centrality, for example; they may also have some built-in analytical support, where, for example, Cypher [Francis et al., 2018] allows for finding shortest paths. In the other direction, analytics can contribute to the querying process in terms of optimisations, where, for example, analysis of connectivity may suggest how to better distribute a large data graph over multiple machines for querying using, e.g., minimum cuts [Akhter et al., 2018, Janke et al., 2018]. Analytics have also been used to rank query results over large graphs [Wagner et al., 2012, Fan et al., 2013], selecting the most important results for presentation to the user.

In some use-cases we may further wish to interleave querying and analytical processes. For example, from the full data graph collected by the tourist board, consider an upcoming airline strike where the board wishes to find the events during the strike with venues in cities unreachable from Santiago by public transport due to the strike. Hypothetically, we could use a query to extract the transport network excluding the airline’s routes (assuming, per Figure 2.3a that the airline information is available), use analytics to extract the strongly connected component containing Santiago, and finally use a query to find events in cities not in the Santiago component on the given dates.21note 21 Such a task could not be solved in a single query using regular path queries as such expressions would not be capable of filtering edges representing flights of a particular airline. While one could solve this task using an imperative language such as Gremlin [Rodriguez, 2015], GraphX [Xin et al., 2013a], or R [The R Foundation, 1992], more declarative languages are also being explored to express such tasks, with proposals including the extension of graph query languages with recursive capabilities [Bischof et al., 2012, Reutter et al., 2015, Hogan et al., 2020],22note 22 Recursive query languages become Turing complete assuming one can also express operations on binary arrays. combining linear algebra with relational (query) algebra [Hutchison et al., 2017], and so forth.

Analytics with entailment

Knowledge graphs are often associated with a semantic schema or ontology that defines the semantics of domain terms, giving rise to entailments (per Chapter 4). Applying analytics with or without such entailments – e.g., before or after materialisation – may yield radically different results. For example, observe that an edge Santa LucíahostsEID15 is semantically equivalent to an edge EID15venueSanta Lucía once the inverse axiom hostsinv. ofvenue is invoked; however, these edges are far from equivalent from the perspective of analytical techniques that consider edge direction, for which including one type of edge, or the other, or both, may have a major bearing on the final results. To the best of our knowledge, the combination of analytics and entailment has not been well-explored, leaving open interesting research questions. Along these lines, it may be of interest to explore semantically-invariant analytics that yield the same results over semantically-equivalent graphs (i.e., graphs that entail one another), thus analysing the semantic content of the knowledge graph rather than simply the topological features of the data graph; for example, semantically-invariant analytics would yield the same results over a graph containing the inverse axiom hostsinv. ofvenue and a number of hosts edges, the same graph but where every hosts edge is replaced by an inverse venue edge, and the union of both graphs.

Knowledge Graph Embeddings

Methods for machine learning have gained significant attention in recent years. In the context of knowledge graphs, machine learning can either be used for directly refining a knowledge graph [Paulheim, 2017] (discussed further in Chapter 8); or for downstream tasks using the knowledge graph, such as recommendation [Zhang et al., 2016], information extraction [Vashishth et al., 2018], question answering [Huang et al., 2019], query relaxation [Wang et al., 2018], query approximation [Hamilton et al., 2018], etc. (discussed further in Chapter 10). However, many traditional machine learning techniques assume dense numeric input representations in the form of vectors, which is quite distinct from how graphs are usually expressed. So how can graphs – or nodes, edges, etc., thereof – be encoded as numeric vectors?

A first attempt to represent a graph using vectors would be to use a one-hot encoding, generating a vector for each node of length \(|L| \cdot |V|\) – with \(|V|\) the number of nodes in the input graph and \(|L|\) the number of edge labels – placing a one at the corresponding index to indicate the existence of the respective edge in the graph, or zero otherwise. Such a representation will, however, typically result in large and sparse vectors, which will be detrimental for most machine learning models.

The main goal of knowledge graph embedding techniques is to create a dense representation of the graph (i.e., embed the graph) in a continuous, low-dimensional vector space that can then be used for machine learning tasks. The dimensionality \(d\) of the embedding is fixed and usually low (often, e.g., \(50 \geq d \geq 1000\)). Typically the graph embedding is composed of an entity embedding for each node: a vector with \(d\) dimensions that we denote by \(\mathbf{e}\); and a relation embedding for each edge label: (typically) a vector with \(d\) dimensions that we denote by \(\mathbf{r}\). The overall goal of these vectors is to abstract and preserve latent structures in the graph. There are many ways in which this notion of an embedding can be instantiated. Most commonly, given an edge spo, a specific embedding approach defines a scoring function that accepts \(\mathbf{e}\)s (the entity embedding of node s), \(\mathbf{r}\)p (the entity embedding of edge label p) and \(\mathbf{e}\)o (the entity embedding of node o) and computes the plausibility of the edge, which estimates how likely it is to be true. Given a data graph, the goal is then to compute the embeddings of dimension \(d\) that maximise the plausibility of positive edges (typically edges in the graph) and minimise the plausibility of negative examples (typically edges in the graph with a node or edge label changed such that they are no longer in the graph) according to the given scoring function. The resulting embeddings can then be seen as models learnt through self-supervision that encode (latent) features of the graph, mapping input edges to output plausibility scores.

Embeddings can then be used for a number of low-level tasks involving the nodes and edge-labels of the graph from which they were computed. First, we can use the plausibility scoring function to assign a confidence to edges that may, for example, have been extracted from an external source (discussed later in Chapter 6). Second, the plausibility scoring function can be used to complete edges with missing nodes/edge labels for the purposes of link prediction (discussed later in Chapter 8); for example, in Figure 5.2, we might ask which nodes in the graph are likely to complete the edge Grey Glacierbus?, where – aside from Punta Arenas, which is already given – we might intuitively expect Torres del Paine to be a plausible candidate. Third, embedding models will typically assign similar vectors to similar nodes and similar edge-labels, and thus they can be used as the basis of similarity measures, which may be useful for finding duplicate nodes that refer to the same entity, or for the purposes of providing recommendations (discussed later in Chapter 10).

A wide range of knowledge graph embedding techniques have been proposed [Wang et al., 2017]. Our goal here is to provide a high-level introduction to some of the most popular techniques proposed thus far. We first discuss tensor-based approaches that include three different sub-approaches using linear/tensor algebra to compute embeddings. We then discuss language models that leverage existing word embedding techniques, proposing ways of generating graph-like analogues for their expected (textual) inputs. Finally we discuss entailment-aware models that can take into account the semantics of the graph, when available.

Tensor-based models

We first discuss tensor-based models, which we sub-divide into three categories: translational models that adopt a geometric perspective whereby relation embeddings translate subject entities to object entities, tensor decomposition models that extract latent factors approximating the graph’s structure, and neural models that use neural networks to train embeddings that provide accurate plausibility scores.

Translational models

Translational models interpret edge labels as transformations from subject nodes (aka the source or head) to object nodes (aka the target or tail); for example, in the edge San PedrobusMoon Valley, the edge label bus is seen as transforming San Pedro to Moon Valley, and likewise for other bus edges. The most elementary approach in this family is TransE [Bordes et al., 2013]. Over all positive edges spo, TransE learns vectors \(\mathbf{e}\)s, \(\mathbf{r}\)p, and \(\mathbf{e}\)os aiming to make \(\mathbf{e}\)s + \(\mathbf{r}\)p as close as possible to \(\mathbf{e}\)o. Conversely, if the edge is a negative example, TransE attempts to learn a representation that keeps \(\mathbf{e}\)s + \(\mathbf{r}\)p away from \(\mathbf{e}\)o. To illustrate, Figure 5.5 provides a toy example of two-dimensional (\(d = 2\)) entity and relation embeddings computed by TransE. We keep the orientation of the vectors similar to the original graph for clarity. For any edge spo in the original graph, adding the vectors \(\mathbf{e}\)s + \(\mathbf{r}\)p should approximate \(\mathbf{e}\)o. In this toy example, the vectors correspond precisely where, for instance, adding the vectors for Licantén (\(\mathbf{e}\)L.) and west of (\(\mathbf{r}\)wo.) gives a vector corresponding to Curico (\(\mathbf{e}\)C.). We can use these embeddings to predict edges (amongst other tasks); for example, in order to predict which node in the graph is most likely to be west of Antofagasta (A.), by computing \(\mathbf{e}\)A. + \(\mathbf{r}\)wo. we find that the resulting vector (dotted in Figure 5.5c) is closest to \(\mathbf{e}\)T., thus predicting Toconao (T.) to be the most plausible such node.

Original graph
Original graph
Relation embeddings
Relation embeddings
Link prediction
Entity embeddings
Toy example of two-dimensional relation and entity embeddings learnt by TransE; the entity embeddings use abbreviations and include an example of vector addition to predict what is west of Antofagasta

Aside from this toy example, TransE can be too simplistic; for example, in Figure 5.2, bus not only transforms San Pedro to Moon Valley, but also to Arica, Calama, and so forth. TransE will, in this case, aim to give similar vectors to all such target locations, which may not be feasible given other edges. TransE will also tend to assign cyclical relations a zero vector, as the directional components will tend to cancel each other out. To resolve such issues, many variants of TransE have been investigated. Amongst these, for example, TransH [Wang et al., 2014] represents different relations using distinct hyperplanes, where for the edge spo, s is first projected onto the hyperplane of p before the translation to o is learnt (uninfluenced by edges with other labels for s and for o). TransR [Lin et al., 2015] generalises this approach by projecting s and o into a vector space specific to p, which involves multiplying the entity embeddings for s and o by a projection matrix specific to p. TransD [Ji et al., 2015] simplifies TransR by associating entities and relations with a second vector, where these secondary vectors are used to project the entity into a relation-specific vector space. Recently, RotatE [Sun et al., 2019] proposes translational embeddings in complex space, which allows to capture more characteristics of relations, such as direction, symmetry, inversion, antisymmetry, and composition. Embeddings have also been proposed in non-Euclidean; for example, MuRP [Balazevic et al., 2019a] uses relation embeddings that transform entity embeddings in the hyperbolic space of the Poincaré ball mode, whose curvature provides more “space” to separate entities with respect to the dimensionality. For discussion of other translational models, we refer to surveys by Cai et al. [2018], Wang et al. [2017].

Tensor decomposition models

A second approach to derive graph embeddings is to apply methods based on tensor decomposition. A tensor is a multidimensional numeric field that generalises scalars (\(0\)-order tensors), vectors (\(1\)-order tensors) and matrices (\(2\)-order tensors) towards arbitrary dimension/order. Tensors have become a widely used abstraction for machine learning [Rabanser et al., 2017]. Tensor decomposition involves decomposing a tensor into more “elemental” tensors (e.g., of lower order) from which the original tensor can be recomposed (or approximated) by a fixed sequence of basic operations over the output tensors. These elemental tensors can be viewed as capturing latent factors underlying the information contained in the original tensor. There are many approaches to tensor decomposition, where we will now briefly introduce the main ideas behind rank decompositions [Rabanser et al., 2017].

Leaving aside graphs momentarily, consider an \((a,b)\)-matrix (i.e., a \(2\)-order tensor) \(\mathbf{C}\), where \(a\) is the number of cities in Chile, \(b\) is the number of months in a year, and each element \((\mathbf{C})_{ij}\) denotes the average temperature of the \(i\)th city in the \(j\)th month. Noting that Chile is a long, thin country – ranging from subpolar climates in the south, to a desert climate in the north – we may find a decomposition of \(\mathbf{C}\) into two vectors representing latent factors – specifically \(\mathbf{x}\) (with \(a\) elements) giving lower values for cities with lower latitude, and \(\mathbf{y}\) (with \(b\) elements), giving lower values for months with lower temperatures – such that computing the outer product23note 23 The outer product of two (column) vectors \(\mathbf{x}\) of length \(a\) and \(\mathbf{y}\) of length \(b\), denoted \(\mathbf{x} \otimes \mathbf{y}\), is defined as \(\mathbf{x}\mathbf{y}^{\mathrm{T}}\), yielding an \((a,b)\)-matrix \(\mathbf{M}\) such that \((\mathbf{M})_{ij} = (\mathbf{x})_i \cdot (\mathbf{y})_j\). Analogously, the outer product of \(k\) vectors is a \(k\)-order tensor. of the two vectors approximates \(\mathbf{C}\) reasonably well: \(\mathbf{x} \otimes \mathbf{y} \approx \mathbf{C}\). In the (unlikely) case that there exist vectors \(\mathbf{x}\) and \(\mathbf{y}\) such that \(\mathbf{C}\) is precisely the outer product of two vectors (\(\mathbf{x} \otimes \mathbf{y} = \mathbf{C}\)) we call \(\mathbf{C}\) a rank-\(1\) matrix; we can then precisely encode \(\mathbf{C}\) using \(a + b\) values rather than \(a \times b\) values. Most times, however, to get precisely \(\mathbf{C}\), we need to sum multiple rank-\(1\) matrices, where the rank \(r\) of \(\mathbf{C}\) is the minimum number of rank-\(1\) matrices that need to be summed to derive precisely \(\mathbf{C}\), such that \(\mathbf{x}_1 \otimes \mathbf{y}_1 + \ldots \mathbf{x}_r \otimes \mathbf{y}_r = \mathbf{C}\). In the temperature example, \(\mathbf{x}_2 \otimes \mathbf{y}_2\) might correspond to a correction for altitude, \(\mathbf{x}_3 \otimes \mathbf{y}_3\) for higher temperature variance further south, etc. A (low) rank decomposition of a matrix then sets a limit \(d\) on the rank and computes the vectors \((\mathbf{x}_1,\mathbf{y}_1,\ldots,\mathbf{x}_{d},\mathbf{y}_{d})\) such that \(\mathbf{x}_1 \otimes \mathbf{y}_1 + \ldots + \mathbf{x}_{d} \otimes \mathbf{y}_{d}\) gives the best \(d\)-rank approximation of \(\mathbf{C}\). Noting that to generate \(n\)-order tensors we need to compute the outer product of \(n\) vectors, we can generalise this idea towards low-rank decomposition of tensors; this method is called Canonical Polyadic (CP) decomposition [Hitchcock, 1927]. For example, a \(3\)-order tensor \(\mathcal{C}\) containing monthly temperatures for Chilean cities at four different times of day could be approximated with \(\mathbf{x}_1 \otimes \mathbf{y}_1 \otimes \mathbf{z}_1 + \ldots \mathbf{x}_{d} \otimes \mathbf{y}_{d} \otimes \mathbf{z}_{d}\) (e.g., \(\mathbf{x}_1\) might be a latitude factor, \(\mathbf{y}_1\) a monthly variation factor, and \(\mathbf{z}_1\) a daily variation factor, and so on). Various algorithms exist to compute (approximate) CP decompositions, including Alternating Least Squares, Jennrich’s Algorithm, and the Tensor Power method [Rabanser et al., 2017].

Returning to graphs, similar principles can be used to decompose a graph into vectors, thus yielding embeddings. In particular, a graph can be encoded as a one-hot \(3\)-order tensor \(\mathcal{G}\) with \(|V| \times |L| \times |V|\) elements, where the element \((\mathcal{G})_{ijk}\) is set to one if the \(i\)th node links to the \(k\)th node with an edge having the \(j\)th label, or zero otherwise. As previously mentioned, such a tensor will typically be very large and sparse, where rank decompositions are thus applicable. A CP decomposition [Hitchcock, 1927] would compute a sequence of vectors \((\mathbf{x}_1,\mathbf{y}_1,\mathbf{z}_1,\ldots,\mathbf{x}_d,\mathbf{y}_d,\mathbf{z}_d)\) such that \(\mathbf{x}_1 \otimes \mathbf{y}_1 \otimes \mathbf{z}_1 + \ldots + \mathbf{x}_d \otimes \mathbf{y}_d \otimes \mathbf{z}_d \approx \mathcal{G}\). We illustrate this scheme in Figure 5.6. Letting \(\mathbf{X}, \mathbf{Y}, \mathbf{Z}\) denote the matrices formed by \(\begin{bmatrix} \mathbf{x}_1\,\cdots\,\mathbf{x}_d \end{bmatrix}\), \(\begin{bmatrix} \mathbf{y}_1\,\cdots\,\mathbf{y}_d \end{bmatrix}\), \(\begin{bmatrix} \mathbf{z}_1\,\cdots\,\mathbf{z}_d \end{bmatrix}\), respectively, with each vector forming a column of the corresponding matrix, we could then extract the \(i\)th row of \(\mathbf{Y}\) as an embedding for the \(i\)th relation, and the \(j\)th rows of \(\mathbf{X}\) and \(\mathbf{Z}\) as two embeddings for the \(j\)th entity. However, knowledge graph embeddings typically aim to assign one vector to each entity.

Abstract illustration of a CP \(d\)-rank decomposition of a tensor representing the graph of Figure 27a
Abstract illustration of a CP \(d\)-rank decomposition of a tensor representing the graph of Figure 5.5a

DistMult [Yang et al., 2015] is a seminal method for computing knowledge graph embeddings based on rank decompositions, where each entity and relation is associated with a vector of dimension \(d\), such that for an edge spo, a plausibility scoring function \(\sum_{i=1}^d (\mathbf{e}\)s\()_i (\mathbf{r}\)p\()_i (\mathbf{e}\)o\()_i\) is defined, where \((\mathbf{e}\)s\()_i\), \((\mathbf{r}\)p\()_i\) and \((\mathbf{e}\)o\()_i\) denote the \(i\)th elements of vectors \(\mathbf{e}\)s, \(\mathbf{r}\)p, \(\mathbf{e}\)o, respectively. The goal, then, is to learn vectors for each node and edge label that maximise the plausibility of positive edges and minimise the plausibility of negative edges. This approach equates to a CP decomposition of the graph tensor \(\mathcal{G}\), but where entities have one vector that is used twice: \(\mathbf{x}_1 \otimes \mathbf{y}_1 \otimes \mathbf{x}_1 + \ldots + \mathbf{x}_d \otimes \mathbf{y}_d \otimes \mathbf{x}_d \approx \mathcal{G}\). A weakness of this approach is that per the scoring function, the plausibility of spo will always be equal to that of ops; in other words, DistMult does not consider edge direction.

Rather than use a vector as a relation embedding, RESCAL [Nickel and Tresp, 2013] uses a matrix, which allows for combining values from \(\mathbf{e}\)s and \(\mathbf{e}\)o across all dimensions, and thus can capture (e.g.) edge direction. However, RESCAL incurs a higher cost in terms of space and time than DistMult. HolE [Nickel et al., 2016b] uses vectors for relation and entity embeddings, but proposes to use the circular correlation operator – which takes sums along the diagonals of the outer product of two vectors – to combine them. This operator is not commutative, and can thus consider edge direction. ComplEx [Trouillon et al., 2016], on the other hand, uses a complex vector (i.e., a vector containing complex numbers) as a relational embedding, which similarly allows for breaking the aforementioned symmetry of DistMult’s scoring function while keeping the number of parameters low. SimplE [Kazemi and Poole, 2018] rather proposes to compute a standard CP decomposition computing two initial vectors for entities from \(\mathbf{X}\) and \(\mathbf{Z}\) and then averaging terms across \(\mathbf{X}\), \(\mathbf{Y}\), \(\mathbf{Z}\) to compute the final plausibility scores. TuckER [Balazevic et al., 2019b] employs a different type of decomposition – called a Tucker Decomposition [Tucker, 1964], which computes a smaller “core” tensor \(\mathcal{T}\) and a sequence of three matrices \(\mathbf{A}\), \(\mathbf{B}\) and \(\mathbf{C}\), such that \(\mathcal{G} \approx \mathcal{T} \otimes \mathbf{A} \otimes \mathbf{B} \otimes \mathbf{C}\) – where entity embeddings are taken from \(\mathbf{A}\) and \(\mathbf{C}\), while relation embeddings are taken from \(\mathbf{B}\). Of these approaches, TuckER [Balazevic et al., 2019b] currently provides state-of-the-art results on standard benchmarks.

Neural models

A limitation of the aforementioned approaches is that they assume either linear (preserving addition and scalar multiplication) or bilinear (e.g., matrix multiplication) operations over embeddings to compute plausibility scores. Other approaches rather use neural networks to learn embeddings with non-linear scoring functions for plausibility.

One of the earliest proposals of a neural model was Semantic Matching Energy (SME) [Glorot et al., 2013], which learns parameters (aka weights: \(\mathbf{w}\), \(\mathbf{w}'\)) for two functions – \(f_{\mathbf{w}}(\mathbf{e}\)s\(,\mathbf{r}\)p\()\) and \(g_{\mathbf{w}'}(\mathbf{e}\)o\(,\mathbf{r}\)p\()\) – such that the dot product of the result of both functions – \(f_{\mathbf{w}}(\mathbf{e}\)s\(,\mathbf{r}\)p\() \cdot g_{\mathbf{w}'}(\mathbf{e}\)o\(,\mathbf{r}\)p\()\) – gives the plausibility score. Both linear and bilinear variants of \(f_{\mathbf{w}}\) and \(g_{\mathbf{w}'}\) are proposed. Another early proposal was Neural Tensor Networks (NTN) [Socher et al., 2013], which proposes to maintain a tensor \(\mathcal{W}\) of internal weights, such that the plausibility score is computed by a complex function that combines the outer product \(\mathbf{e}\)s\( \otimes \mathcal{W} \otimes \mathbf{e}\)o with a standard neural layer over \(\mathbf{e}\)s and \(\mathbf{e}\)o, which in turn is combined with \(\mathbf{r}\)p, to produce a plausibility score. The tensor \(\mathcal{W}\) results in a high number of parameters, limiting scalability [Wang et al., 2017]. Multi-Layer Perceptron (MLP) [Dong et al., 2014] is a simpler model, where \(\mathbf{e}\)s, \(\mathbf{r}\)p and \(\mathbf{e}\)o are concatenated and fed into a hidden layer to compute plausibility scores.

A number of more recent approaches have proposed using convolutional kernels in their models. ConvE [Dettmers et al., 2018] proposes to generate a matrix from \(\mathbf{e}\)s and \(\mathbf{r}\)p by “wrapping” each vector over several rows and concatenating both matrices. The concatenated matrix serves as the input for a set of (2D) convolutional layers, which returns a feature map tensor. The feature map tensor is vectorised and projected into \(d\) dimensions using a parameterised linear transformation. The plausibility score is then computed based on the dot product of this vector and \(\mathbf{e}\)o. A disadvantage of ConvE is that by wrapping vectors into matrices, it imposes an artificial two-dimensional structure on the embeddings. HypER [Balazevic et al., 2019c] is a similar model using convolutions, but avoids the need to wrap vectors into matrices. Instead, a fully connected layer (called the “hypernetwork”) is applied to \(\mathbf{r}\)p and used to generate a matrix of relation-specific convolutional filters. These filters are applied directly to \(\mathbf{e}\)s to give a feature map, which is vectorised. The same process is then applied as in ConvE: the resulting vector is projected into \(d\) dimensions, and a dot product applied with \(\mathbf{e}\)o to produce the plausibility score. The resulting model is shown to outperform ConvE on standard benchmarks [Balazevic et al., 2019c].

The presented approaches strike different balances in terms of expressivity and the number of parameters than need to be trained. While more expressive models, such as NTN, may better fit more complex plausibility functions over lower dimensional embeddings by using more hidden parameters, simpler models, such as that proposed by Dong et al. [Dong et al., 2014], and convolutional networks [Dettmers et al., 2018, Balazevic et al., 2019c] that enable parameter sharing by applying the same (typically small) kernels over different regions of a matrix, require handling fewer parameters overall and are more scalable.

Survey and definition of tensor-based approaches

We now formally define and survey the aforementioned tensor-based approaches. For simplicity, we will consider directed edge-labelled graphs.

Before defining embeddings, we first introduce tensors.

Vector, matrix, tensor, order, mode
For any positive integer \(a\), a vector of dimension \(a\) is a family of real numbers indexed by integers in \(\{1, \ldots, a\}\). For \(a\) and \(b\) positive integers, an \((a,b)\)-matrix is a family of real numbers indexed by pairs of integers in \(\{1, \ldots, a\} \times \{1, \ldots, b\}\). A tensor is a family of real numbers indexed by a finite sequence of integers such that there exist positive numbers \(a_1, \ldots, a_n\) such that the indices are all the tuples of numbers in \(\{1, \ldots, a_1\} \times \ldots \times \{1, \ldots, a_n\}\). The number \(n\) is called the order of the tensor, the subindices \(i\in \{1, \ldots, n\}\) indicate the mode of a tensor, and each \(a_i\) defines the dimension of the \(i\)th mode. A 1-order tensor is a vector and a 2-order tensor is a matrix. We denote the set of all tensors as \(\mathbb{T}\).

For specific dimensions \(a_1,\ldots,a_n\) of modes, a tensor is an element of \((\cdots(\mathbb{R}^{a_1})^{\ldots})^{a_n}\) but we write \(\mathbb{R}^{a_1,\ldots,a_n}\) to simplify the notation. We use lower-case bold font to denote vectors (\(\mathbf{x} \in \mathbb{R}^a\)), upper-case bold font to denote matrices (\(\mathbf{X} \in \mathbb{R}^{a,b}\)) and calligraphic font to denote tensors (\(\mathcal{X} \in \mathbb{R}^{a_1,\ldots,a_n}\)).

Now we are ready to abstractly define knowledge graph embeddings.

Knowledge graph embedding
Given a directed edge-labelled graph \(G = (V,E,L)\), a knowledge graph embedding of \(G\) is a pair of mappings \((\varepsilon,\rho)\) such that \(\varepsilon : V \rightarrow \mathbb{T}\) and \(\rho : L \rightarrow \mathbb{T}\).

In the most typical case, \(\varepsilon\) and \(\rho\) map nodes and edge-labels, respectively, to vectors of fixed dimension. In some cases, however, they may map to matrices. Given this abstract notion of a knowledge graph embedding, we can then define a plausibility scoring function.

Plausibility scores
A plausibility scoring function is a partial function \(\phi : \mathbb{T} \times \mathbb{T} \times \mathbb{T} \rightarrow \mathbb{R}\). Given a directed edge-labelled graph \(G = (V,E,L)\), an edge \((s,p,o) \in V \times L \times V\), and a knowledge graph embedding \((\varepsilon,\rho)\) of \(G\), the plausibility of \((s,p,o)\) is given as \(\phi(\varepsilon(s),\rho(p),\varepsilon(o))\).

Edges with higher scores are considered more plausible. Given a graph \(G = (V,E,L)\), we assume a set of positive edges \(E^+\) and a set of negative edges \(E^{-}\). Positive edges are often simply the edges in the graph: \(E^+ \coloneqq E\). Negative edges use the vocabulary of \(G\) (i.e., \(E^- \subseteq V \times L \times V\)) and are typically defined by taking edges \((s,p,o)\) from \(E\) and changing one term of each edge – often one of the nodes – such that the edge is no longer in \(E\). Given sets of positive and negative edges, and a plausibility scoring function, the objective is then to find the embedding that maximises the plausibility of edges in \(E^+\) while minimising the plausibility of edges in \(E^{-}\). Specific knowledge graph embeddings then instantiate the type of embedding considered and the plausibility scoring function in various ways.

In Table 5.1, we define the plausibility scoring function and types of embeddings used by different knowledge graph embeddings. To simplify the definitions, we use \(\mathbf{e}_x\) to denote \(\varepsilon(x)\) when it is a vector, \(\mathbf{r}_y\) to denote \(\rho(y)\) when it is a vector, and \(\mathbf{R}_y\) to denote \(\rho(y)\) when it is a matrix. Some models involve learnt parameters (aka weights) for computing plausibility. We denote these as \(\mathbf{v}\), \(\mathbf{V}\), \(\mathcal{V}\), \(\mathbf{w}\), \(\mathbf{W}\) \(\mathcal{W}\) (for vectors, matrices or tensors). We use \(d_e\) and \(d_r\) to denote the dimensionality chosen for entity embeddings and relation embeddings, respectively. Often it is assumed that \(d_e = d_r\), in which case we will write \(d\). Weights may have their own dimensionality, which we denote \(w\). The embeddings in Table 5.1 use a variety of operators on vectors, matrices and tensors, which will be defined later.

The embeddings defined in Table 5.1 vary in complexity, where a trade-off exists between the number of parameters used, and the expressiveness of the model in terms of its capability to capture latent features of the graph. To increase expressivity, many of the models in Table 5.1 use additional parameters beyond the embeddings themselves. A possible formal guarantee of such models is full expressiveness, which, given any disjoint sets of positive edges \(E^+\) and negative edges \(E^{-}\), asserts that the model can always correctly partition those edges. On the one hand, for example, DistMult [Yang et al., 2015] cannot distinguish an edge spo from its inverse ops, so by adding an inverse of an edge in \(E^+\) to \(E^{-}\), we can show that it is not fully expressive. On the other hand, models such as ComplEx [Trouillon et al., 2016], SimplE [Kazemi and Poole, 2018], and TuckER [Balazevic et al., 2019b] have been proven to be fully expressive given sufficient dimensionality; for example, TuckER [Balazevic et al., 2019b] with dimensions \(d_r = |L|\) and \(d_e = |V|\) trivially satisfies full expressivity since its core tensor \(\mathcal{W}\) then has sufficient capacity to store the full one-hot encoding of any graph. This formal property is useful to show that the model does not have built-in limitations for numerically representing a graph, though of course in practice the dimensions needed to reach full expressivity are often impractical/undesirable.

We continue by first defining the conventions used in Table 5.1.

  • We use \((\mathbf{x})_{i}\), \((\mathbf{X})_{ij}\), and \((\mathcal{X})_{{i_1}\ldots{i_n}}\) to denote elements of vectors, matrices, and tensors, respectively. If a vector \(\mathbf{x} \in \mathbb{R}^a\) is used in a context that requires a matrix, the vector is interpreted as an \((a, 1)\)-matrix (i.e., a column vector) and can be turned into a row vector (i.e., a \((1,a)\)-matrix) using the transpose operation \(\mathbf{x}^T\). We use \(\mathbf{x}^\mathrm{D} \in \mathbb{R}^{a,a}\) to denote the diagonal matrix with the values of the vector \(\mathbf{x} \in \mathbb{R}^{a}\) on its diagonal. We denote the identity matrix by \(\mathbf{I}\) such that if \(j=k\), then \((\mathbf{I})_{jk} = 1\); otherwise \((\mathbf{I})_{jk} = 0\).
  • We denote by \(\begin{bmatrix}\mathbf{X}_1\\[-0.5ex]\vdots\\\mathbf{X_n}\end{bmatrix}\) the vertical stacking of matrices \(\mathbf{X}_1, \ldots, \mathbf{X}_n\) with the same number of columns. Given a vector \(\mathbf{x} \in \mathbb{R}^{ab}\), we denote by \(\mathbf{x}^{[a,b]} \in \mathbb{R}^{a,b}\) the “reshaping” of \(\mathbf{x}\) into an \((a,b)\)-matrix such that \((\mathbf{x}^{[a,b]})_{ij} = (\mathbf{x})_{(i + a(j-1))}\). Conversely, given a matrix \(\mathbf{X} \in \mathbb{R}^{a,b}\), we denote by \(\mathrm{vec}(\mathbf{X}) \in \mathbb{R}^{ab}\) the vectorisation of \(\mathbf{X}\) such that \(\mathrm{vec}(\mathbf{X})_k = (\mathbf{X})_{ij}\) where \(i = ((k-1)\,\mathrm{mod}\,m) + 1\) and \(j = \frac{k - i}{m} + 1\) (observe that \(\mathrm{vec}(\mathbf{x}^{[a,b]}) = \mathbf{x}\)).
  • Given a tensor \(\mathcal{X} \in \mathbb{R}^{a,b,c}\), we denote by \(\mathcal{X}^{[i:\cdot:\cdot]} \in \mathbb{R}^{b,c}\), the \(i\)th slice of tensor \(\mathcal{X}\) along the first mode; for example, given \(\mathcal{X} \in \mathbb{R}^{5,2,3}\), then \(\mathcal{X}^{[4:\cdot:\cdot]}\) returns the \((2,3)\)-matrix consisting of the elements \(\begin{bmatrix} (\mathcal{X})_{411} & (\mathcal{X})_{412} & (\mathcal{X})_{413} \\ (\mathcal{X})_{421} & (\mathcal{X})_{422} & (\mathcal{X})_{423} \end{bmatrix}\). Analogously, we use \(\mathcal{X}^{[\cdot : i : \cdot]} \in \mathbb{R}^{a,c}\) and \(\mathcal{X}^{[\cdot:\cdot:i]} \in \mathbb{R}^{b,c}\) to indicate the \(i\)th slice along the second and third modes of \(\mathcal{X}\), respectively.
  • We denote by \(\psi(\mathcal{X})\) the element-wise application of a function \(\psi\) to the tensor \(\mathcal{X}\), such that \((\psi(\mathcal{X}))_{in_1\ldots i_n} = \psi(\mathcal{X}_{i_1\ldots i_n})\). Common choices for \(\psi\) include a sigmoid function (e.g., the logistic function \(\psi(x) = \frac{1}{1 + e^{-x}}\) or the hyperbolic tangent function \(\psi(x) = \mathrm{tanh}\,x = \frac{e^x - e^{-x}}{e^x + e^{-x}}\)), the rectifier (\(\psi(x) = \mathrm{max}(0,x)\)), softplus (\(\psi(x) = \mathrm{ln}(1 + e^x)\)), etc.

We now define the operators used in Table 5.1, where the first and most elemental operation we consider is that of matrix multiplication.

Matrix multiplication
The multiplication of matrices \(\mathbf{X} \in \mathbb{R}^{a,b}\) and \(\mathbf{Y} \in \mathbb{R}^{b,c}\) is a matrix \(\mathbf{XY} \in \mathbb{R}^{a,c}\) such that \((\mathbf{XY})_{ij} = \sum_{k=1}^b (\mathbf{X})_{ik}(\mathbf{Y})_{kj}\). The matrix multiplication of two tensors \(\mathcal{X} \in \mathbb{R}^{a_1,\ldots,a_m,c}\) and \(\mathcal{Y} \in \mathbb{R}^{c,b_1,\ldots,b_n}\) is a tensor \(\mathcal{XY} \in \mathbb{R}^{a_1,\ldots,a_{m},b_{1},\ldots,b_{n}}\) such that (\(\mathcal{XY})_{i_1\ldots i_m i_{m+1}\ldots i_{m+n}} = \sum_{k=1}^c (\mathcal{X})_{i_1\ldots i_m k}(\mathcal{Y})_{k i_{m+1}i_{m+n}}\).

For convenience, we may implicitly add or remove modes with dimension 1 for the purposes of matrix multiplication and other operators; for example, given two vectors \(\mathbf{x} \in \mathbb{R}^{a}\) and \(\mathbf{y} \in \mathbb{R}^{a}\), we denote by \(\T{\mathbf{x}}\mathbf{y}\) (aka the dot or inner product) the multiplication of matrix \(\T{\mathbf{x}} \in \mathbb{R}^{1,a}\) with \(\mathbf{y} \in \mathbb{R}^{a,1}\) such that \(\T{\mathbf{x}}\mathbf{y} \in \mathbb{R}^{1,1}\) (i.e., a scalar in \(\mathbb{R}\)); conversely, \(\mathbf{x}\T{\mathbf{y}} \in \mathbb{R}^{a,a}\) (the outer product).

Constraints on embeddings are sometimes given as norms, defined next.

\(L^p\)-norm, \(L^{p,q}\)-norm
For \(p\in \mathbb{R}\), the \(L^p\)-norm of a vector \(\mathbf{x}\in \mathbb{R}^a\) is the scalar \(\|\mathbf{x}\|_p \coloneqq (|(\mathbf{x})_1|^p + \ldots + |(\mathbf{x})_a|^p)^{\frac{1}{p}}\), where \(|(\mathbf{x})_i|\) denotes the absolute value of the \(i\)th element of \(\mathbf{x}\). For \(p,q\in \mathbb{R}\), the \(L^{p,q}\)-norm of a matrix \(\mathbf{X}\in\mathbb{R}^{a,b}\) is the scalar \(\|\mathbf{X}\|_{p,q} \coloneqq \left( \sum_{j=1}^b \left( \sum_{i=1}^a |(\mathbf{X})_{ij}|^p \right)^{\frac{q}{p}} \right)^\frac{1}{q}\).

The \(L^1\) norm (i.e., \(\|\mathbf{x}\|_1\)) is thus simply the sum of the absolute values of \(\mathbf{x}\), while the \(L^2\) norm (i.e., \(\|\mathbf{x}\|_2\)) is the (Euclidean) length of the vector. The Frobenius norm of the matrix \(\mathbf{X}\) then equates to \(\|\mathbf{X}\|_{2,2} = \left( \sum_{j=1}^b \left( \sum_{i=1}^a |(\mathbf{X})_{ij}|^2 \right) \right)^\frac{1}{2}\); i.e., the square root of the sum of the squares of all elements.

Another type of product used by embedding techniques is the Hadamard product, which multiplies tensors of the same dimension and computes their product in an element-wise manner.

Hadamard product
Given two tensors \(\mathcal{X} \in \mathbb{R}^{a_1,\ldots,a_n}\) and \(\mathcal{Y} \in \mathbb{R}^{a_1,\ldots,a_n}\), the Hadamard product \(\mathcal{X} \odot \mathcal{Y}\) is defined as a tensor in \(\mathbb{R}^{a_1,\ldots,a_n}\), with each element computed as \((\mathcal{X} \odot \mathcal{Y})_{i_1\ldots i_{n}} \coloneqq (\mathcal{X})_{i_1\ldots i_{n}} (\mathcal{Y})_{i_1\ldots i_{n}}\).

Other embedding techniques – namely RotatE [Sun et al., 2019] and ComplEx [Trouillon et al., 2016] – uses complex space based on complex numbers. With a slight abuse of notation, the definitions of vectors, matrices and tensors can be modified by replacing the set of real numbers \(\mathbb{R}\) by the set of complex numbers \(\mathbb{C}\), giving rise to complex vectors, complex matrices, and complex tensors. In this case, we denote by \(\mathrm{Re}(\cdot)\) the real part of a complex number. Given a complex vector \(\mathbf{x} \in \mathbb{C}^I\), we denote by \(\overline{\mathbf{x}}\) its complex conjugate (swapping the sign of the imaginary part of each element). Complex analogues of the aforementioned operators can then be defined by replacing the multiplication and addition of real numbers with the analogous operators for complex numbers, where RotateE [Sun et al., 2019] uses the complex Hadamard product, and ComplEx [Trouillon et al., 2016] uses complex matrix multiplication.

One embedding technique – MuRP [Balazevic et al., 2019a] – uses hyperbolic space, specifically based on the Poincaré ball. As this is the only embedding we cover that uses this space, and the formalisms are lengthy (covering the Poincaré ball, Möbius addition, Möbius matrix–vector multiplication, logarithmic maps, exponential maps, etc.), we rather refer the reader to the paper for further details [Balazevic et al., 2019a].

As discussed in Section 5.2, tensor decompositions are used for many embeddings, and at the heart of such decompositions is the tensor product, which is often used to reconstruct (an approximation of) the original tensor.

Tensor product
Given two tensors \(\mathcal{X} \in \mathbb{R}^{a_1,\ldots,a_m}\) and \(\mathcal{Y} \in \mathbb{R}^{b_1,\ldots,b_n}\), the tensor product \(\mathcal{X} \otimes \mathcal{Y}\) is defined as a tensor in \(\mathbb{R}^{a_1,\ldots,a_m,b_1,\ldots,b_n}\), with each element computed as \((\mathcal{X} \otimes \mathcal{Y})_{i_1\ldots i_{m}j_1\ldots j_n} \coloneqq (\mathcal{X})_{i_1 \ldots i_m} (\mathcal{Y})_{j_1 \ldots j_n}\).24note 24 Please note that “\(\otimes\)” is used here in an unrelated sense to its use in Definition 3.10.

Assume that \(\mathcal{X} \in \mathbb{R}^{2,3}\) and \(\mathcal{Y} \in \mathbb{R}^{3,4,5}\). Then \(\mathcal{X} \otimes \mathcal{Y}\) will be a tensor in \(\mathbb{R}^{2,3,3,4,5}\). Element \((\mathcal{X} \otimes \mathcal{Y})_{12345}\) will be the product of \((\mathcal{X})_{12}\) and \((\mathcal{Y})_{345}\).

An \(n\)-mode product is used by other embeddings to transform elements along a given mode of a tensor by computing a product with a given matrix along that particular mode of the tensor.

\(n\)-mode product
For a positive integer \(n\), a tensor \(\mathcal{X} \in \mathbb{R}^{a_1,\ldots,a_{n-1},a_n,a_{n+1},\ldots,a_m}\) and matrix \(\mathbf{Y} \in \mathbb{R}^{b,a_n}\), the \(n\)-mode product of \(\mathcal{X}\) and \(\mathbf{Y}\) is the tensor \(\mathcal{X} \otimes_n \mathbf{Y} \in \mathbb{R}^{a_1,\ldots,a_{n-1},b,a_{n+1},\ldots,a_m}\) such that \((\mathcal{X} \otimes_n \mathbf{Y})_{i_1\ldots i_{n-1}ji_{n+1}\ldots i_m} \coloneqq \sum_{k=1}^{a_n} (\mathcal{X})_{i_1 \ldots i_{n-1}ki_{n+1} \ldots i_m} (\mathbf{Y})_{jk}\).

Let us assume that \(\mathcal{X} \in \mathbb{R}^{2,3,4}\) and \(\mathbf{Y} \in \mathbb{R}^{5,3}\). The result of \(\mathcal{X} \otimes_2 \mathbf{Y}\) will be a tensor in \(\mathbb{R}^{2,5,4}\), where, for example, \((\mathcal{X} \otimes_2 \mathbf{Y})_{142}\) will be given as \((\mathcal{X})_{112}(\mathbf{Y})_{41} + (\mathcal{X})_{122}(\mathbf{Y})_{42} + (\mathcal{X})_{132}(\mathbf{Y})_{43}\). Observe that if \(\mathbf{y} \in \mathbb{R}^{a_n}\) – i.e., if \(\mathbf{y}\) is a (column) vector – then the \(n\)-mode tensor product \(\mathcal{X} \otimes_n \T{\mathbf{y}}\) “flattens” the \(n\)th mode of \(\mathcal{X}\) to one dimension, effectively reducing the order of \(\mathcal{X}\) by one.

One embedding technique – HolE [Nickel et al., 2016b] – uses the circular correlation operator \(\mathbf{x} \star \mathbf{y}\), where each element is the sum of elements along a diagonal of the outer product \(\mathbf{x} \otimes \mathbf{y}\) that “wraps” if not the primary diagonal.

Circular correlation
The circular correlation of vector \(\mathbf{x} \in \mathbb{R}^a\) with \(\mathbf{y} \in \mathbb{R}^a\) is the vector \(\mathbf{x} \star \mathbf{y} \in \mathbb{R}^{a}\) such that \((\mathbf{x} \star \mathbf{y})_k \coloneqq \sum_{i=1}^a (\mathbf{x})_i (\mathbf{y})_{(((k+i-2) \,\mathrm{mod}\,a)+1)}\).

Assuming \(a = 5\), then \((\mathbf{x} \star \mathbf{y})_1 = (\mathbf{x})_1(\mathbf{y})_1 + (\mathbf{x})_2(\mathbf{y})_2 + (\mathbf{x})_3(\mathbf{y})_3 + (\mathbf{x})_4(\mathbf{y})_4 + (\mathbf{x})_5(\mathbf{y})_5\), or a case that wraps: \((\mathbf{x} \star \mathbf{y})_4 = (\mathbf{x})_1(\mathbf{y})_4 + (\mathbf{x})_2(\mathbf{y})_5 + (\mathbf{x})_3(\mathbf{y})_1 + (\mathbf{x})_4(\mathbf{y})_2 + (\mathbf{x})_5(\mathbf{y})_3\).

Finally, a couple of neural models that we include – namely ConvE [Dettmers et al., 2018] and HypER [Balazevic et al., 2019c] – are based on convolutional architectures using the convolution operator.

Convolution
Given two matrices \(\mathbf{X} \in \mathbb{R}^{a,b}\) and \(\mathbf{Y} \in \mathbb{R}^{e,f}\), the convolution of \(\mathbf{X}\) and \(\mathbf{Y}\) is the matrix \(\mathbf{X} * \mathbf{Y} \in \mathbb{R}^{(a + e - 1),(b + f - 1)}\) such that \((\mathbf{X} * \mathbf{Y})_{ij} = \sum_{k=1}^a \sum_{l=1}^b (\mathbf{X})_{kl} (\mathbf{Y})_{(i+k-a)(j+l-b)}\).25note 25 We define the convolution operator per the widely-usedconvention for convolutional neural networks. Strictly speaking, the operator should be called cross-correlation, where traditional convolution requires the matrix \(\mathbf{X}\) to be initially “rotated” by 180°. Since in our settings the matrix \(\mathbf{X}\) is learnt, rather than given, the rotation is redundant, and hence the distinction is not important. In cases where \((i+k-a) < 1\), \((j+l-b) < 1\), \((i+k-a) > e\) or \((j+l-b) > f\) (i.e., where \((\mathbf{Y})_{(i+k-a)(j+l-b)}\) lies outside the bounds of \(\mathbf{Y}\)), we say that \((\mathbf{Y})_{(i+k-a)(j+l-b)} = 0\).

Intuitively speaking, the convolution operator overlays \(\mathbf{X}\) in every possible way over \(\mathbf{Y}\) such that at least one pair of elements \((\mathbf{X})_{ij},(\mathbf{Y})_{lk}\) overlaps, summing the products of pairs of overlapping elements to generate an element of the result. Elements of \(\mathbf{X}\) extending beyond \(\mathbf{Y}\) are ignored (equivalently we can consider \(\mathbf{Y}\) to be “zero-padded” outside its borders).

Given \(\mathbf{X} \in \mathbb{R}^{3,3}\) and \(\mathbf{Y} \in \mathbb{R}^{4,5}\), then \(\mathbf{X} * \mathbf{Y} \in \mathbb{R}^{6,7}\), where, for example, \((\mathbf{X} * \mathbf{Y})_{11} = (\mathbf{X})_{33}(\mathbf{Y})_{11}\) (with the bottom right corner of \(\mathbf{X}\) overlapping the top left corner of \(\mathbf{Y}\)), while \((\mathbf{X} * \mathbf{Y})_{34} = (\mathbf{X})_{11}(\mathbf{Y})_{12} + \)\( (\mathbf{X})_{12}(\mathbf{Y})_{13} + \)\( (\mathbf{X})_{13}(\mathbf{Y})_{14} + \)\( (\mathbf{X})_{21}(\mathbf{Y})_{22} + \)\( (\mathbf{X})_{22}(\mathbf{Y})_{23} + \)\( (\mathbf{X})_{23}(\mathbf{Y})_{24} + \)\( (\mathbf{X})_{31}(\mathbf{Y})_{32} + \)\( (\mathbf{X})_{32}(\mathbf{Y})_{33} + \)\( (\mathbf{X})_{33}(\mathbf{Y})_{34}\) (with \((\mathbf{X})_{22}\) – the centre of \(\mathbf{X}\) – overlapping \((\mathbf{Y})_{23}\)).26note 26 Models applying convolutions may differ regarding how edge cases are handled, or on the “stride” of the convolution applied, where, for example, a stride of 3 for \((\mathbf{X} * \mathbf{Y})\) would see the kernel \(\mathbf{X}\) centred only on elements \((\mathbf{Y})_{ij}\) such that \(i\,\mathrm{mod}\,3 = 0\) and \(j\,\mathrm{mod}\,3 = 0\), reducing the number of output elements by a factor of 9. We do not consider such details here.

In a convolution \(\mathbf{X} * \mathbf{Y}\), the matrix \(\mathbf{X}\) is often called the “kernel” (or “filter”). Often several kernels are used in order to apply multiple convolutions. Given a tensor \(\mathcal{X} \in \mathbb{R}^{c,a,b}\) (representing \(c\) \((a,b)\)-kernels) and a matrix \(\mathbf{Y} \in \mathbb{R}^{e,f}\), we denote by \(\mathcal{X} * \mathbf{Y} \in \mathbb{R}^{c,(a + e - 1),(b + f - 1)}\) the result of the convolutions of the \(c\) first-mode slices of \(\mathcal{X}\) over \(\mathbf{Y}\) such that \((\mathcal{X} * \mathbf{Y})^{[i:\cdot:\cdot]} = \mathcal{X}^{[i:\cdot:\cdot]} * \mathbf{Y}\) for \(1 \leq i \leq c\), yielding a tensor of results for \(c\) convolutions.

Details for selected knowledge graph embeddings, including the plausibility scoring function \(\phi(\varepsilon(s),\rho(p),\varepsilon(o))\) for edge \(s\)\(p\)\(o\), and other conditions
Model \(\phi(\varepsilon(s),\rho(p),\varepsilon(o))\) Conditions (for all \(x \in V\), \(y \in L\))
TransE \(- \|\mathbf{e}_s + \mathbf{r}_p - \mathbf{e}_o\|_q\) \(\mathbf{e}_x \in \mathbb{R}^{d}\), \(\mathbf{r}_y \in \mathbb{R}^{d}\), \(q \in \{1,2\}\), \(\|\mathbf{e}_x\|_2 = 1\)
TransH \(-\|(\mathbf{e}_s - (\T{\mathbf{e}_s}\mathbf{w}_p)\mathbf{w}_p) + \mathbf{r}_p - (\mathbf{e}_o - (\T{\mathbf{e}_o} \mathbf{w}_p)\mathbf{w}_p)\|^{2}_{2}\) \(\mathbf{e}_x \in \mathbb{R}^{d}\), \(\mathbf{r}_y \in \mathbb{R}^{d}\), \(\mathbf{w}_y \in \mathbb{R}^d\),
\(\|\mathbf{w}_y\|_2 = 1\) , \(\frac{\T{\mathbf{w}_y} \mathbf{r}_y}{\|\mathbf{r}_y\|_2} \approx 0\), \(\|\mathbf{e}_x\|_2 \leq 1\)
TransR \(-\|\mathbf{W}_p\mathbf{e}_s + \mathbf{r}_p - \mathbf{W}_p\mathbf{e}_o\|^{2}_{2}\) \(\mathbf{e}_x \in \mathbb{R}^{d_e}\), \(\mathbf{r}_y \in \mathbb{R}^{d_r}\), \(\mathbf{W}_y \in \mathbb{R}^{d_r , d_e}\),
\(\|\mathbf{e}_x\|_2 \leq 1\), \(\|\mathbf{r}_y\|_2 \leq 1\), \(\|\mathbf{W}_y\mathbf{e}_x\|_2 \leq 1\)
TransD \(-\|(\mathbf{w}_p\otimes\mathbf{w}_s + \mathbf{I})\mathbf{e}_s + \mathbf{r}_p - (\mathbf{w}_p\otimes\mathbf{w}_o + \mathbf{I})\mathbf{e}_o\|^{2}_{2}\) \(\mathbf{e}_x \in \mathbb{R}^{d_e}\), \(\mathbf{r}_y \in \mathbb{R}^{d_r}\), \(\mathbf{w}_x \in \mathbb{R}^{d_e}\), \(\mathbf{w}_y \in \mathbb{R}^{d_r}\),
\(\|\mathbf{e}_x\|_2 \leq 1\), \(\|\mathbf{r}_y\|_2 \leq 1\), \(\|(\mathbf{w}_y\otimes\mathbf{w}_x + \mathbf{I})\mathbf{e}_x\|_2 \leq 1\)
RotatE \(- \|\mathbf{e}_s \odot \mathbf{r}_p - \mathbf{e}_o\|_2\) \(\mathbf{e}_x \in \mathbb{C}^{d}\), \(\mathbf{r}_y \in \mathbb{C}^{d}\), \(\|\mathbf{r}_y\|_2 = 1\)
RESCAL \(\T{\mathbf{e}_s} \mathbf{R}_p \mathbf{e}_o\) \(\mathbf{e}_x \in \mathbb{R}^{d}\), \(\mathbf{R}_y \in \mathbb{R}^{d,d}\), \(\|\mathbf{e}_x\|_2 \leq 1\), \(\|\mathbf{R}_y\|_{2,2} \leq 1\)
DistMult \(\T{\mathbf{e}_s} \D{\mathbf{r}_p} \mathbf{e}_o\) \(\mathbf{e}_x \in \mathbb{R}^{d}\), \(\mathbf{r}_y \in \mathbb{R}^{d}\), \(\|\mathbf{e}_x\|_2 = 1\), \(\|\mathbf{r}_y\|_2 \leq 1\)
HolE \(\T{\mathbf{r}_p} (\mathbf{e}_s \star \mathbf{e}_o)\) \(\mathbf{e}_x \in \mathbb{R}^{d}\), \(\mathbf{r}_y \in \mathbb{R}^{d}\), \(\|\mathbf{e}_x\|_2 \leq 1\), \(\|\mathbf{r}_y\|_2 \leq 1\)
ComplEx \(\mathrm{Re}(\T{\mathbf{e}_s} \D{\mathbf{r}_p} \overline{\mathbf{e}}_o)\) \(\mathbf{e}_x \in \mathbb{C}^{d}\), \(\mathbf{r}_y \in \mathbb{C}^{d}\), \(\|\mathbf{e}_x\|_2 \leq 1\), \(\|\mathbf{r}_y\|_2 \leq 1\)
SimplE \(\frac{\T{\mathbf{e}_s} \D{\mathbf{r}_p} \mathbf{w}_o + \T{\mathbf{e}_o} \D{\mathbf{w}_p} \mathbf{w}_s}{2}\) \(\mathbf{e}_x \in \mathbb{R}^{d}\), \(\mathbf{r}_y \in \mathbb{R}^{d}\), \(\mathbf{w}_x \in \mathbb{R}^{d}\), \(\mathbf{w}_y \in \mathbb{R}^{d}\),
\(\|\mathbf{e}_x\|_2 \leq 1\), \(\|\mathbf{w}_x\|_2 \leq 1\), \(\|\mathbf{r}_y\|_2 \leq 1, \|\mathbf{w}_y\|_2 \leq 1\)
TuckER \(\mathcal{W} \otimes_1 \T{\mathbf{e}_s} \otimes_2 \T{\mathbf{r}_p} \otimes_3 \T{\mathbf{e}_o}\) \(\mathbf{e}_x \in \mathbb{R}^{d_e}\), \(\mathbf{r}_y \in \mathbb{R}^{d_r}\), \(\mathcal{W} \in \mathbb{R}^{d_e , d_r , d_e }\)
SME L. \(\T{(\mathbf{V}\mathbf{e}_s + \mathbf{V}'\mathbf{r}_p + \mathbf{v})} (\mathbf{W}\mathbf{e}_o + \mathbf{W}'\mathbf{r}_p + \mathbf{w})\) \(\mathbf{e}_x \in \mathbb{R}^{d}\), \(\mathbf{r}_y \in \mathbb{R}^{d}\), \(\mathbf{v} \in \mathbb{R}^w\), \(\mathbf{w} \in \mathbb{R}^w\), \(\|\mathbf{e}_x\|_2 = 1\),
\(\mathbf{V} \in \mathbb{R}^{w,d},\mathbf{V}' \in \mathbb{R}^{w,d}, \mathbf{W} \in \mathbb{R}^{w,d}, \mathbf{W}' \in \mathbb{R}^{w,d}\)
SME Bi. \(\T{((\mathcal{V} \otimes_3 \T{\mathbf{r}_p}) \mathbf{e}_s + \mathbf{v})}((\mathcal{W} \otimes_3 \T{\mathbf{r}_p}) \mathbf{e}_o + \mathbf{w})\) \(\mathbf{e}_x \in \mathbb{R}^{d}\), \(\mathbf{r}_y \in \mathbb{R}^{d}\), \(\mathbf{v} \in \mathbb{R}^w\), \(\mathbf{w} \in \mathbb{R}^w\), \(\|\mathbf{e}_x\|_2 = 1\),
\(\mathcal{V} \in \mathbb{R}^{w,d,d}\), \(\mathcal{W} \in \mathbb{R}^{w,d,d}\)
NTN \(\T{\mathbf{r}_p} \psi\left(\T{\mathbf{e}_s} \mathcal{W} \mathbf{e}_o + \mathbf{W} \begin{bmatrix}\mathbf{e}_s\\\mathbf{e}_o\end{bmatrix} + \mathbf{w}\right) \) \(\mathbf{e}_x \in \mathbb{R}^{d}\), \(\mathbf{r}_y \in \mathbb{R}^{d}\), \(\mathbf{w} \in \mathbb{R}^{w}\), \(\mathbf{W} \in \mathbb{R}^{w , 2d}\),
\(\mathcal{W} \in \mathbb{R}^{d,w,d}\), \(\|\mathbf{e}_x\|_2 \leq 1\), \(\|\mathbf{r}_y\|_2 \leq 1\),
\(\|\mathbf{w}\|_2 \leq 1\), \(\|\mathbf{W}\|_{2,2} \leq 1\), \(\|\mathcal{W}^{[\cdot:i:\cdot]}_{1\leq i \leq w}\|_{2,2} \leq 1\)
MLP \(\T{\mathbf{v}} \psi\left(\mathbf{W} \begin{bmatrix}\mathbf{e}_s\\\mathbf{r}_p\\\mathbf{e}_o\end{bmatrix} + \mathbf{w}\right) \) \(\mathbf{e}_x \in \mathbb{R}^{d}\), \(\mathbf{r}_y \in \mathbb{R}^{d}\), \(\mathbf{v} \in \mathbb{R}^{w}\), \(\mathbf{w} \in \mathbb{R}^{w}\), \(\mathbf{W} \in \mathbb{R}^{w , 3d}\)
\(\|\mathbf{e}_x\|_2 \leq 1\) \(\|\mathbf{r}_y\|_2 \leq 1\)
ConvE \(\psi\left(\T{\mathrm{vec}\left(\psi\left( \mathcal{W} * \begin{bmatrix}\mathbf{e}_s^{[a, b]}\\\mathbf{r}_p^{[a, b]}\end{bmatrix} \right)\right)} \mathbf{W}\right) \mathbf{e}_o \) \(\mathbf{e}_x \in \mathbb{R}^{d}\), \(\mathbf{r}_y \in \mathbb{R}^{d}\), \(d = ab\),
\(\mathbf{W} \in \mathbb{R}^{w_1(w_2 + 2a - 1)(w_3 + b - 1) , d}\), \(\mathcal{W} \in \mathbb{R}^{w_1 , w_2 , w_3}\)
HypER \(\psi\T{\left(\mathrm{vec}\left( \T{\mathbf{r}_p} \mathcal{W} * \mathbf{e}_s \right)} \mathbf{W} \right) \mathbf{e}_o\) \(\mathbf{e}_x \in \mathbb{R}^{d_e}\), \(\mathbf{r}_y \in \mathbb{R}^{d_r}\), \(\mathbf{W} \in \mathbb{R}^{w_2(w_1 + d_e - 1) , d_e}\),
\(\mathcal{W} \in \mathbb{R}^{d_r , w_1 , w_2}\)

Language models

Embedding techniques were first explored as a way to represent natural language within machine learning frameworks, with word2vec [Mikolov et al., 2013] and GloVe [Pennington et al., 2014] being two seminal approaches. Both approaches compute embeddings for words based on large corpora of text such that words used in similar contexts (e.g., “frog”, “toad”) have similar vectors. Word2vec uses neural networks trained either to predict the current word from surrounding words (continuous bag of words), or to predict the surrounding words given the current word (continuous skip-gram). GloVe rather applies a regression model over a matrix of co-occurrence probabilities of word pairs. Embeddings generated by both approaches have become widely used in natural language processing tasks.

Another approach for graph embeddings is thus to leverage proven approaches for language embeddings. However, while a graph consists of an unordered set of sequences of three terms (i.e., a set of edges), text in natural language consists of arbitrary-length sequences of terms (i.e., sentences of words). RDF2Vec [Ristoski and Paulheim, 2016] thus performs (biased [Cochez et al., 2017a]) random walks on the graph and records the paths (the sequence of nodes and edge labels traversed) as “sentences”, which are then fed as input into the word2vec [Mikolov et al., 2013] model. An example of such a path extracted from Figure 5.2 might be, for example, San PedrobusCalamaflightIquiqueflightSantiago, where the paper experiments with \(500\) paths of length \(8\) per entity. RDF2Vec also proposes a second mode where sequences are generated for nodes from canonically-labelled sub-trees of which they are a root node, where sub-trees of depth \(1\) and \(2\) are used for experiments. KGloVe [Cochez et al., 2017b] is rather based on GloVe. Given that the original GloVe model [Pennington et al., 2014] considers words that co-occur frequently in windows of text to be more related, KGloVe uses personalised PageRank27note 27 Intuitively speaking, personalised PageRank starts at a given node and then determines the probability of a random walk being at a particular node after a given number of steps. A higher number of steps converges towards standard PageRank emphasising global node centrality in the graph, while a lower number emphasises proximity/relatedness to the starting node. to determine the most related nodes to a given node, which are fed into the GloVe model.

Entailment-aware models

The embeddings thus far consider the data graph alone. But what if an ontology or set of rules is provided? Such deductive knowledge could be used to improve the embeddings. One approach is to use constraint rules to refine the predictions made by embeddings; for example, Wang et al. [2015] use functional and inverse-functional definitions as constraints (under UNA) such that, for example, if we define that an event can have at most one value for venue, this is used to lower the plausibility of edges that would assign multiple venues to an event.

More recent approaches rather propose joint embeddings that consider both the data graph and rules when computing embeddings. KALE [Guo et al., 2016] computes entity and relation embeddings using a translational model (specifically TransE) that is adapted to further consider rules using t-norm fuzzy logics. With reference to Figure 5.2, consider a simple rule ?xbus?y \(\Rightarrow\) ?xconnects to?y. We can use embeddings to assign plausibility scores to new edges, such as \(e_1\): Piedras RojasbusMoon Valley. We can further apply the previous rule to generate a new edge \(e_2\): Piedras Rojasconnects toMoon Valley from the predicted edge \(e_1\). But what plausibility should we assign to this second edge? Letting \(p_1\) and \(p_2\) be the current plausibility scores of \(e_1\) and \(e_2\) (initialised using the standard embedding), then t-norm fuzzy logics suggests that the plausibility be updated as \(p_1p_2 - p_1 + 1\). Embeddings are then trained to jointly assign larger plausibility scores to positive examples versus negative examples of both edges and ground rules. An example of a positive ground rule based on Figure 5.2 would be AricabusSan Pedro \(\Rightarrow\) Aricaconnects toSan Pedro. Negative ground rules randomly replace the relation in the head of the rule; for example, AricabusSan Pedro \(\not\Rightarrow\) AricaflightSan Pedro. Guo et al. [2018] later propose RUGE, which uses a joint model over ground rules (possibly soft rules with confidence scores) and plausibility scores to align both forms of scoring for unseen edges.

Generating ground rules can be costly. An alternative approach, called FSL [Demeester et al., 2016], observes that in the case of a simple rule, such as ?xbus?y \(\Rightarrow\) ?xconnects to?y, the relation embedding bus should always return a lower plausibility than connects to. Thus, for all such rules, FSL proposes to train relation embeddings while avoiding violations of such inequalities. While relatively straightforward, FSL only supports simple rules, while KALE also supports more complex rules.

These works exemplify how deductive and inductive forms of knowledge – in this case rules and embeddings – can interplay and complement each other.

Graph Neural Networks

While embeddings aim to provide a dense numerical representation of graphs suitable for use within existing machine learning models, another approach is to build custom machine learning models adapted for graph-structured data. Most custom learning models for graphs are based on (artificial) neural networks [Wu et al., 2019], exploiting a natural correspondence between both: a neural network already corresponds to a weighted, directed graph, where nodes serve as artificial neurons, and edges serve as weighted connections (axons). However, the typical topology of a traditional neural network – more specifically, a fully-connected feed-forward neural network – is quite homogeneous, being defined in terms of sequential layers of nodes where each node in one layer is connected to all nodes in the next layer. Conversely, the topology of a data graph is quite heterogeneous, being determined by the relations between entities that its edges represent.

A graph neural network (GNN) [Scarselli et al., 2009] builds a neural network based on the topology of the data graph; i.e., nodes are connected to their neighbours per the data graph. Typically a model is then learnt to map input features for nodes to output features in a supervised manner; output features of the example nodes used for training may be manually labelled, or may be taken from the knowledge graph. Unlike knowledge graph embeddings, GNNs support end-to-end supervised learning for specific tasks: given a set of labelled examples, GNNs can be used to classify elements of the graph or the graph itself. GNNs have been used to perform classification over graphs encoding compounds, objects in images, documents, etc.; as well as to predict traffic, build recommender systems, verify software, etc. [Wu et al., 2019]. Given labelled examples, GNNs can even replace graph algorithms; for example, GNNs have been used to find central nodes in knowledge graphs in a supervised manner [Scarselli et al., 2009, Park et al., 2019, Park et al., 2020].

We now discuss the ideas underlying two main flavours of GNN, specifically, recursive GNNs and non-recursive GNNs.

Recursive graph neural networks

Recursive graph neural networks (RecGNNs) are the seminal approach to graph neural networks [Sperduti and Starita, 1997, Scarselli et al., 2009]. The approach is conceptually similar to the systolic abstraction illustrated in Figure 5.3, where messages are passed between neighbours towards recursively computing some result. However, rather than define the functions used to decide the messages to pass, we rather label the output of a training set of nodes and let the framework learn the functions that generate the expected output, thereafter applying them to label other examples.

In a seminal paper, Scarselli et al. [2009] proposed what they generically call a graph neural network (GNN), which takes as input a directed graph where nodes and edges are associated with feature vectors that can capture node and edge labels, weights, etc. These feature vectors remain fixed throughout the process. Each node in the graph is also associated with a state vector, which is recursively updated based on information from the node’s neighbours – i.e., the feature and state vectors of the neighbouring nodes and the feature vectors of the edges extending to/from them – using a parametric function, called the transition function. A second parametric function, called the output function, is used to compute the final output for a node based on its own feature and state vector. These functions are applied recursively up to a fixpoint. Both parametric functions can be implemented using neural networks where, given a partial set of supervised nodes in the graph – i.e., nodes labelled with their desired output – parameters for the transition and output functions can be learnt that best approximate the supervised outputs. The result can thus be seen as a recursive neural network architecture.28note 28 Some authors refer to such architectures as recurrent graph neural networks, observing that the internal state maintained for nodes can be viewed as a form of recurrence over a sequence of transitions. To ensure convergence up to a fixpoint, certain restrictions are applied, namely that the transition function be a contractor, meaning that upon each application of the function, points in the numeric space are brought closer together (intuitively, in this case, the numeric space “shrinks” upon each application, ensuring convergence to a unique fixpoint).

To illustrate, consider, for example, that we wish to find priority locations for creating new tourist information offices. A good strategy would be to install them in hubs from which many tourists visit popular destinations. Along these lines, in Figure 5.7 we illustrate the GNN architecture proposed by Scarselli et al. [2009] for a sub-graph of Figure 5.2, where we highlight the neighbourhood of Punta Arenas. In this graph, nodes are annotated with feature vectors (\(\mathbf{n}_x\)) and hidden states at step \(t\) (\(\mathbf{h}_x^{(t)}\)), while edges are annotated with feature vectors (\(\mathbf{a}_{xy}\)). Feature vectors for nodes may, for example, one-hot encode the type of node (City, Attraction, etc.), directly encode statistics such as the number of tourists visiting per year, etc. Feature vectors for edges may, for example, one-hot encode the edge label (the type of transport), directly encode statistics such as the distance or number of tickets sold per year, etc. Hidden states can be randomly initialised. The right-hand side of Figure 5.7 provides the GNN transition and output functions, where \(\mathrm{N}(x)\) denotes the neighbouring nodes of \(x\), \(f_{\mathbf{w}}(\cdot)\) denotes the transition function with parameters \(\mathbf{w}\), and \(g_{\mathbf{w}'}(\cdot)\) denotes the output function with parameters \(\mathbf{w'}\). An example is also provided for Punta Arenas (\(x = 1\)). These functions will be recursively applied until a fixpoint is reached. To train the network, we can label examples of places that already have (or should have) tourist offices and places that do (or should) not have tourist offices. These labels may be taken from the knowledge graph, or may be added manually. The GNN can then learn parameters \(\mathbf{w}\) and \(\mathbf{w'}\) that give the expected output for the labelled examples, which can subsequently be used to label other nodes.

This GNN model is flexible and can be adapted in various ways [Scarselli et al., 2009]: we may define neighbouring nodes differently, for example to include nodes for outgoing edges, or nodes one or two hops away; we may allow pairs of nodes to be connected by multiple edges with different vectors; we may consider transition and output functions with distinct parameters for each node; we may add states and outputs for edges; we may change the sum to another aggregation function; etc.

On the left, a sub-graph of Figure 24 highlighting the neighbourhood of Punta Arenas, where nodes are annotated with feature vectors (n_x) and hidden states at step t (h_x^{(t)}), and edges are annotated with feature vectors (a_{xy}); on the right, the GNN transition and output functions proposed by Scarselli et al. and an example for Punta Arenas (x = 1), where N(x) denotes the neighbouring nodes of x, f_w(·)\) denotes the transition function with parameters w and g_{w'}(·) denotes the output function with parameters w'
 
\(\mathbf{h}_x^{(t)} \coloneqq\) \(\sum_{y \in \textrm{N}(x)} f_\mathbf{w}(\mathbf{n}_{x},\mathbf{n}_{y},\mathbf{a}_{yx},\mathbf{h}_{y}^{(t-1)})\)
\(\mathbf{o}_x^{(t)} \coloneqq\) \(g_{\mathbf{w}'}(\mathbf{h}_x^{(t)},\mathbf{n}_x)\)
\(\mathbf{h}_1^{(t)} \coloneqq\) \(f_\mathbf{w}(\mathbf{n}_{1},\mathbf{n}_{3},\mathbf{a}_{31},\mathbf{h}_{3}^{(t-1)})\)
\(+ f_\mathbf{w}(\mathbf{n}_{1},\mathbf{n}_{4},\mathbf{a}_{41},\mathbf{h}_{4}^{(t-1)})\)
\(\mathbf{o}_1^{(t)} \coloneqq\) \(g_{\mathbf{w}'}(\mathbf{h}_1^{(t)},\mathbf{n}_1)\)
\(\ldots\)
 
On the left a sub-graph of Figure 5.2 highlighting the neighbourhood of Punta Arenas, where nodes are annotated with feature vectors (\(\mathbf{n}_x\)) and hidden states at step \(t\) (\(\mathbf{h}_x^{(t)}\)), and edges are annotated with feature vectors (\(\mathbf{a}_{xy}\)); on the right, the GNN transition and output functions proposed by Scarselli et al. [2009] and an example for Punta Arenas (\(x = 1\)), where \(\mathrm{N}(x)\) denotes the neighbouring nodes of \(x\), \(f_{\mathbf{w}}(\cdot)\) denotes the transition function with parameters \(\mathbf{w}\) and \(g_{\mathbf{w}'}(\cdot)\) denotes the output function with parameters \(\mathbf{w'}\)

We now define a recursive graph neural network. We assume that the GNN accepts a directed vector-labelled graph as input (see Definition 5.1).

Recursive graph neural network
A recursive graph neural network (RecGNN) is a pair of functions \(\mathfrak{R} \coloneqq (\)Agg, Out\()\), such that (with \(a, b, c \in \mathbb{N}\)):
  • Agg\(: \mathbb{R}^a \times 2^{(\mathbb{R}^a \times \mathbb{R}^b) \rightarrow \mathbb{N}} \rightarrow \mathbb{R}^a\)
  • Out\(: \mathbb{R}^a \rightarrow \mathbb{R}^c\)

The function Agg computes a new feature vector for a node, given its previous feature vector and the feature vectors of the nodes and edges forming its neighbourhood; the function Out transforms the final feature vector computed by Agg for a node to the output vector for that node. We assume that \(a\) and \(b\) correspond to the dimensions of the input node and edge vectors, respectively, while \(c\) denotes the dimension of the output vector for each node. Given a RecGNN \(\mathfrak{R} = (\)Agg, Out\()\), a directed vector-labelled graph \(G = (V,E,F,\lambda)\), and a node \(u \in V\), we define the output vector assigned to node \(u\) in \(G\) by \(\mathfrak{R}\) (written \(\mathfrak{R}(G,u)\)) as follows. First let \(\mathbf{n}_u^{(0)} \coloneqq \lambda(u)\). For all \(i \geq 1\), let:

\(\mathbf{n}_u^{(i)} \coloneqq\) Agg \(\left( \mathbf{n}_u^{(i-1)}, \{\!\!\{ (\mathbf{n}_v^{(i-1)},\lambda(v,u)) \mid (v,u) \in E \}\!\!\} \right) \)

If \(j \geq 1\) is an integer such that \(\mathbf{n}_u^{(j)} = \mathbf{n}_u^{(j-1)}\) for all \(u \in V\), then \(\mathfrak{R}(G,u) \coloneqq\) Out\((\mathbf{n}_u^{(j)})\).

In a RecGNN, the same aggregation function (Agg) is applied recursively until a fixpoint is reached, at which point an output function (Out}) creates the final output vector for each node. While in practice RecGNNs will often consider a static feature vector and a dynamic state vector [Scarselli et al., 2009], we can more concisely encode this as one vector, where part may remain static throughout the aggregation process representing input features, and part may be dynamically computed representing the state. In practice, Agg and Out are often based on parametric combinations of vectors, with the parameters learnt based on a sample of output vectors for labelled nodes.

The aggregation function for the GNN of Scarselli et al. [2009] is given as:

Agg\((\mathbf{n}_u,N) \coloneqq \sum_{(\mathbf{n}_v,\mathbf{a}_{vu})\in N}f_{\mathbf{w}}(\mathbf{n}_u,\mathbf{n}_v,\mathbf{a}_{vu})\)

where \(f_{\mathbf{w}}(\cdot)\) is a contraction function with parameters \(\mathbf{w}\). The output function is defined as:

Out\(\left( \mathbf{n}_u \right) \coloneqq g_{\textbf{w}'}(\mathbf{n}_u)\)

where again \(g_{\mathbf{w}'}(\cdot)\) is a function with parameters \(\mathbf{w'}\). Given a set of nodes labelled with their expected output vectors, the parameters \(\mathbf{w}\) and \(\mathbf{w}'\) are learnt.

There are notable similarities between graph parallel frameworks (GPFs; see Definition 5.2) and RecGNNs. While we defined GPFs using separate Msg and Agg functions, this is not essential: conceptually they could be defined in a similar way to RecGNN, with a single Agg function that “pulls” information from its neighbours (we maintain Msg to more closely reflect how GPFs are defined/implemented in practice). The key difference between GPFs and GNNs is that in the former, the functions are defined by the user, while in the latter, the functions are generally learnt from labelled examples. Another difference arises from the termination condition present in GPFs, though often the GPF’s termination condition will – like in RecGNNs – reflect convergence to a fixpoint.

Non-recursive graph neural networks

GNNs can also be defined in a non-recursive manner, where a fixed number of layers are applied over the input in order to generate the output. A benefit of this approach is that we do not need to worry about convergence since the process is non-recursive. Also, each layer will often have independent parameters, representing different transformation steps. Naively, a downside is that adding many layers could give rise to a high number of parameters. Addressing this problem, a popular approach for non-recursive GNNs is to use convolutional neural networks.

Convolutional neural networks (CNNs) have gained a lot of attention, in particular, for machine learning tasks involving images [Krizhevsky et al., 2017]. The core idea in the image setting is to train and apply small kernels (aka filters) over localised regions of an image using a convolution operator to extract features from that local region. When applied to all local regions, the convolution outputs a feature map of the image. Since the kernels are small, and are applied multiple times to different regions of the input, the number of parameters to train is reduced. Typically multiple kernels can thus be applied, forming multiple convolutional layers.

One may note that in GNNs and CNNs, operators are applied over local regions of the input data. In the case of GNNs, the transition function is applied over a node and its neighbours in the graph. In the case of CNNs, the convolution is applied on a pixel and its neighbours in the image. Following this intuition, a number of convolutional graph neural networks (ConvGNNs) [Bruna et al., 2014, Kipf and Welling, 2017, Wu et al., 2019] have been proposed, where the transition function is implemented by means of convolutions. A key consideration for ConvGNNs is how regions of a graph are defined. Unlike the pixels of an image, nodes in a graph may have varying numbers of neighbours. This creates a challenge: a benefit of CNNs is that the same kernel can be applied over all the regions of an image, but this requires more careful consideration in the case of ConvGNNs since neighbourhoods of different nodes can be diverse. Approaches to address these challenges involve working with spectral (e.g. [Bruna et al., 2014, Kipf and Welling, 2017]) or spatial (e.g., [Monti et al., 2017]) representations of graphs that induce a more regular structure from the graph. An alternative is to use an attention mechanism [Velickovic et al., 2018] to learn the nodes whose features are most important to the current node.

Next we abstractly define a non-recursive graph neural network.

Non-recursive graph neural network
A non-recursive graph neural network (NRecGNN) with \(l\) layers is an \(l\)-tuple of functions \(\mathfrak{N} \coloneqq (\)Agg\(^{(1)},\ldots,\) Agg\(^{(l)} )\), such that, for \(1 \leq k \leq l\) (with \(a_0, \ldots a_l, b \in \mathbb{N}\)), Agg\(^{(k)}: \mathbb{R}^{a_{k-1}} \times 2^{(\mathbb{R}^{a_{k-1}} \times \mathbb{R}^b) \rightarrow \mathbb{N}} \rightarrow \mathbb{R}^{a_{k}}\).

Each function Agg\(^{(k)}\) (as before) computes a new feature vector for a node, given its previous feature vector and the feature vectors of the nodes and edges forming its neighbourhood. We assume that \(a_0\) and \(b\) correspond to the dimensions of the input node and edge vectors, respectively, where each function Agg\(^{(k)}\) for \(2 \leq k \leq l\) accepts as input node vectors of the same dimension as the output of the function Agg\(^{(k-1)}\). Given an NRecGNN \(\mathfrak{N} = (\) Agg\(^{(1)},\ldots,\) Agg\(^{(l)} )\), a directed vector-labelled graph \(G = (V,E,F,\lambda)\), and a node \(u \in V\), we define the output vector assigned to node \(u\) in \(G\) by \(\mathfrak{N}\) (written \(\mathfrak{N}(G,u)\)) as follows. First let \(\mathbf{n}_u^{(0)} \coloneqq \lambda(u)\). For all \(i \geq 1\), let:

\(\mathbf{n}_u^{(i)} \coloneqq\) Agg\(^{(i)} \left( \mathbf{n}_u^{(i-1)}, \{\!\!\{ (\mathbf{n}_v^{(i-1)},\lambda(v,u)) \mid (v,u) \in E \}\!\!\} \right) \)

Then \(\mathfrak{N}(G,u) \coloneqq \mathbf{n}_u^{(l)}\).

In an \(l\)-layer NRecGNN, a different aggregation function can be applied at each step (i.e., in each layer), up to a fixed number of steps \(l\). We do not consider a separate Out function as it can be combined with the final aggregation function Agg\(^{(l)}\). When the aggregation functions use a convolutional operator based on kernels learned from labelled examples, we call the result a convolutional graph neural network (ConvGNN). We refer to the survey by Wu et al. [2019] for discussion of ConvGNNs proposed in the literature.

We have considered GNNs that define the neighbourhood of a node based on its incoming edges. These definitions can be adapted to also consider outgoing neighbours by either adding inverse edges to the directed vector-labelled graph in pre-processing, or by adding outgoing neighbours as arguments to the Agg\((\cdot)\) function. More generally, GNNs (and indeed GPFs) relying solely on the neighbourhood of each node have limited expressivity in terms of their ability to distinguish nodes and graphs [Xu et al., 2019]; for example, Barceló et al. [2020] show that such NRecGNNs have a similar expressiveness for classifying nodes as the \(\mathcal{ALCQ}\) Description Logic discussed in Section 4.3.2. More expressive GNN variants have been proposed that allow the aggregation functions to access and update a globally shared vector [Barceló et al., 2020]. We refer to the papers by Xu et al. [2019] and Barceló et al. [2020] for further discussion.

Symbolic Learning

The supervised techniques discussed thus far – namely knowledge graph embeddings and graph neural networks – learn numerical models over graphs. However, such models are often difficult to explain or understand. For example, taking the graph of Figure 5.8, knowledge graph embeddings might predict the edge SCLflightARI as being highly plausible, but they will not provide an interpretable model to help understand why this is the case: the reason for the result may lie in a matrix of parameters learnt to fit a plausibility score on training data. Such approaches also suffer from the out-of-vocabulary problem, where they are unable to provide results for edges involving previously unseen nodes or edges; for example, if we add an edge SCLflightCDG, where CDG is new to the graph, a knowledge graph embedding will not have the entity embedding for CDG and would need to be retrained in order to estimate the plausibility of an edge CDGflightSCL.

An incomplete directed edge-labelled graph describing flights between airports
A directed edge-labelled graph describing flights between airports

An alternative (sometimes complementary) approach is to adopt symbolic learning in order to learn hypotheses in a symbolic (logical) language that “explain” a given set of positive and negative edges. These edges are typically generated from the knowledge graph in an automatic manner (similar to the case of knowledge graph embeddings). The hypotheses then serve as interpretable models that can be used for further deductive reasoning. Given the graph of Figure 5.8, we may, for example, learn the rule ?xflight?y \(\Rightarrow\) ?yflight?x from observing that flight routes tend to be return routes. Alternatively, rather than learn rules, we might learn a DL axiom from the graph stating that airports are either domestic, international, or both: Airport \(\sqsubseteq\) DomesticAirport \(\sqcup\) InternationalAirport. Such rules and axioms can then be used for deductive reasoning, and offer an interpretable model for new knowledge that is entailed/predicted; for example, from the aforementioned rule for return flights, one can interpret why a novel edge SCLflightARI is predicted. This further offers domain experts the opportunity to verify the models – e.g., the rules and axioms – derived by such processes. Finally, rules/axioms are quantified (all flights have a return flight, all airports are domestic or international, etc.), so they can be applied to unseen examples (e.g., with the aforementioned rule, we can derive CDGflightSCL from a new edge SCLflightCDG with the unseen node CDG).

In this section, we discuss two forms of symbolic learning: rule mining, which learns rules, and axiom mining, which learns other forms of logical axioms.

Rule mining

Rule mining, in the general sense, refers to discovering meaningful patterns in the form of rules from large collections of background knowledge. In the context of knowledge graphs, we assume a set of positive and negative edges as given. Typically positive edges are observed edges (i.e., those given or entailed by a knowledge graph) while negative edges are defined according to a given assumption of completeness (discussed later). The goal of rule mining is to identify new rules that entail a high ratio of positive edges from other positive edges, but entail a low ratio of negative edges from positive edges. The types of rules considered may vary from more simple cases, such as ?xflight?y \(\Rightarrow\) ?yflight?x mentioned previously, to more complex rules, such as ?xcapital?ynearby?ztypeAirport \(\Rightarrow\) ?ztypeInternational Airport, based on observing in the graph that airports near capitals tend to be international airports; or dom flight rule premise \(\Rightarrow\) ?xdomestic flight?y, indicating that flights within the same country denote domestic flights (as seen previously in Section 4.3.1).

Per the example inferring that airports near capital cities are international airports, rules are not assumed to hold in all cases, but rather are associated with measures of how well they conform to the positive and negative edges. In more detail, we call the edges entailed by a rule and the set of positive edges (not including the entailed edge itself), the positive entailments of that rule. The number of entailments that are positive is called the support for the rule, while the ratio of a rule’s entailments that are positive is called the confidence for the rule [Suchanek et al., 2019]. Support and confidence indicate, respectively, the number and ratio of entailments “confirmed” to be true for the rule, where the goal is to identify rules that have both high support and high confidence. Techniques for rule mining in relational settings have long been explored in the context of Inductive Logic Programming (ILP) [De Raedt, 2008]. However, knowledge graphs present novel challenges due to the scale of the data and the frequent assumption of incomplete data (OWA), where dedicated techniques have been proposed to address these issues [Galárraga et al., 2013].

When dealing with an incomplete knowledge graph, it is not immediately clear how to define negative edges. A common heuristic – also used for knowledge graph embeddings – is to adopt a Partial Completeness Assumption (PCA) [Galárraga et al., 2013], which considers the set of positive edges to be those contained in the data graph, and the set of negative examples to be the set of all edges \(x\)\(p\)\(y'\) not in the graph but where there exists a node \(y\) such that \(x\)\(p\)\(y'\) is in the graph. Taking Figure 5.8, an example of a negative edge under PCA would be SCLflightARI (given the presence of SCLflightLIM); conversely, SCLdomestic flightARI is neither positive nor negative. The PCA confidence measure is then the ratio of the support to the number of entailments in the positive or negative set [Galárraga et al., 2013]. For example, the support for the rule ?xdomestic flight?y \(\Rightarrow\) ?ydomestic flight?x is \(2\) (since it entails IQQdomestic flightARI and ARIdomestic flightIQQ in the graph, which are thus positive edges), while the confidence is \(\frac{2}{2} = 1\) (noting that SCLdomestic flightARI, though entailed, is neither positive nor negative, and is thus ignored by the measure). The support for the rule ?xflight?y \(\Rightarrow\) ?yflight?x is analogously 4, while the confidence is \(\frac{4}{5} = 0.8\) (noting that SCLflightARI is a negative edge).

The goal then, is to find rules satisfying given support and confidence thresholds. An influential rule-mining system for graphs is AMIE [Galárraga et al., 2013, Galárraga et al., 2015], which adopts the PCA measure of confidence, and builds rules in a top-down fashion [Suchanek et al., 2019] starting with rule heads of the form \(\Rightarrow\) ?xcountry?y. For each such rule head (one for each edge label), three types of refinements are considered, each of which adds a new edge to the body of the rule. This new edge takes an edge label from the graph and may otherwise use fresh variables not appearing previously in the rule, existing variables that already appear in the rule, or nodes from the graph. The three refinements may then:

  1. add an edge with one existing variable and one fresh variable; for example, refining the aforementioned rule head might give: ?zflight?x \(\Rightarrow\) ?xcountry?y;
  2. add an edge with an existing variable and a graph node; for example, refining the above rule might give: Domestic Airporttype?zflight?x \(\Rightarrow\) ?xcountry?y;
  3. add an edge with two existing variables; for example, refining the above rule might give: dom airport rule premise \(\Rightarrow\) ?xcountry?y.

These refinements can be combined arbitrarily, which gives rise to a potentially exponential search space, where rules meeting given thresholds for support and confidence are maintained. To improve efficiency, the search space can be pruned; for example, these three refinements always decrease support, so if a rule does not meet the support threshold, there is no need to explore its refinements. Further restrictions are imposed on the types of rules generated. First, only rules up to a certain fixed size are considered. Second, a rule must be closed, meaning that each variable appears in at least two edges of the rule, which ensures that rules are safe, meaning that each variable in the head appears in the body; for example, the rules produced by the first and second refinements in the example are neither closed (variable y appears once) nor safe (variable y appears only in the head).29note 29 Safe rules like ?xcapital?ynearby?ztypeAirport \(\Rightarrow\) ?ztypeInternational Airport are not closed as ?x appears only in one edge. The condition that rules are closed is strictly stronger than the safety condition. The third refinement is thus applied until a rule is closed. For further discussion of possible optimisations based on pruning and indexing, we refer to the paper [Galárraga et al., 2015].

Later works have built on these techniques for mining rules from knowledge graphs. Gad-Elrab et al. [2016] propose a method to learn non-monotonic rules – rules with negated edges in the body – in order to capture exceptions to base rules; for example, the rule not international rule premise \(\Rightarrow\) ?xcountry?y may be learnt, indicating that flights are within the same country except when the (departure) airport is international, where the exception is shown dotted and we use \(\neg\) to negate an edge (representing an exception). The RuLES system [Ho et al., 2018] – which is also capable of learning non-monotonic rules – proposes to mitigate the limitations of the PCA heuristic by extending the confidence measure to consider the plausibility scores of knowledge graph embeddings for entailed edges not appearing in the graph. Where available, explicit statements about the completeness of the knowledge graph (such as expressed in shapes; see Section 3.1.2) can be used in lieu of PCA for identifying negative edges. Along these lines, CARL [Pellissier Tanon et al., 2017] exploits additional knowledge about the cardinalities of relations to refine the set of negative examples and the confidence measure for candidate rules. Alternatively, where available, ontologies can be used to derive logically-certain negative edges under OWA through, for example, disjointness axioms. The system proposed by d’Amato et al. [d'Amato et al., 2016b, d'Amato et al., 2016a] leverages ontologically-entailed negative edges for determining the confidence of rules generated through an evolutionary algorithm.

While the previous works involve discrete expansions of candidate rules for which a fixed confidence scoring function is applied, another line of research is on a technique called differentiable rule mining [Rocktäschel and Riedel, 2017, Yang et al., 2017, Sadeghian et al., 2019], which allows end-to-end learning of rules. The core idea is that the joins in rule bodies can be represented as matrix multiplication. More specifically, we can represent the relations of an edge label \(p\) by the adjacency matrix \(\mathbf{A}_p\) (of size \(|V| \times |V|\)) such that the value on the \(i\)th row of the \(j\)th column is \(1\) if there is an edge labelled \(p\) from the \(i\)th entity to the \(j\)th entity; otherwise the value is \(0\). Now we can represent a join in a rule body as matrix multiplication; for example, given ?xdomestic flight?ycountry?z \(\Rightarrow\) ?xcountry?z, we can denote the body by the matrix multiplication \(\mathbf{A}\)df.\(\mathbf{A}\)c., which gives an adjacency matrix representing entailed country edges, where we should expect the \(1\)’s in \(\mathbf{A}\)df.\(\mathbf{A}\)c. to be covered by the head’s adjacency matrix \(\mathbf{A}\)c.. Since we are given adjacency matrices for all edge labels, we are left to learn confidence scores for individual rules, and to learn rules (of varying length) with a threshold confidence. Along these lines, NeuralLP [Yang et al., 2017] uses an attention mechanism to select a variable-length sequence of edge labels for path-like rules of the form ?xp\(1\)y\(1\)p\(2\)p\(n\)y\(n\)p\(n+1\)?z \(\Rightarrow\) ?xp?z, for which confidences are likewise learnt. DRUM [Sadeghian et al., 2019] also learns path-like rules, where, observing that some edge labels are more/less likely to follow others in the rules – for example, flight will not be followed by capital in the graph of Figure 5.2 as the join will be empty – the system uses bidirectional recurrent neural networks (a popular technique for learning over sequential data) to learn sequences of relations for rules, and their confidences. These differentiable rule mining techniques are, however, currently limited to learning path-like rules.

Axiom mining

More general forms of axioms beyond rules – expressed in logical languages such as DLs (see Section 4.3.2) – can be mined from knowledge graphs. We can divide these approaches into two: those mining specific axioms and more general axioms.

Among systems mining specific types of axioms, disjointness axioms are a popular target; for example, DomesticAirport \(\sqcap\) InternationalAirport \(\equiv \bot\) states that the two classes are disjoint by equivalently stating that the intersection of the two classes is equivalent to the empty class, or in simpler terms, no node can be simultaneously of type Domestic Airport and International Airport. The system proposed by Völker et al. [2015] extracts disjointness axioms based on (negative) association rule mining [Agrawal et al., 1993], which finds pairs of classes where each has many instances in the knowledge graph but there are relatively few (or no) instances of both classes. Töpper et al. [2012] rather extract disjointness for pairs of classes that have a cosine similarity below a fixed threshold. For computing this cosine similarity, class vectors are computed using a TF–IDF analogy, where the “document” of each class is constructed from all of its instances, and the “terms” of this document are the properties used on the class instances (preserving multiplicities). While the previous two approaches find disjointness constraints between named classes (e.g., city is disjoint with airport), Rizzo et al. [2017], Rizzo et al. [2021] propose an approach that can capture disjointness constraints between class descriptions (e.g., city without an airport nearby is disjoint with city that is the capital of a country). The approach first clusters similar nodes of the knowledge base. Next, a terminological cluster tree is extracted, where each leaf node indicates a cluster extracted previously, and each internal (non-leaf) node is a class definition (e.g., cities) where the left child is either a cluster having all nodes in that class or a sub-class description (e.g., cities without airports) and the right child is either a cluster having no nodes in that class or a disjoint-class description (e.g., non-cities with events). Finally, candidate disjointness axioms are proposed for pairs of class descriptions in the tree that are not entailed to have a sub-class relation.

Other systems propose methods to learn more general axioms. One of the first proposals in this direction is the DL-FOIL system [Fanizzi et al., 2008, Rizzo et al., 2020], which is based on algorithms for class learning (aka concept learning), whereby given a set of positive nodes and negative nodes, the goal is to find a logical class description that divides the positive and negative sets. For example, given \(\{\)Iquique, Arica\(\}\) as the positive set and \(\{\)Santiago\(\}\) as the negative set, we may learn a (DL) class description \(\exists\)nearby.Airport \(\sqcap \neg(\exists\) capital\(^-.\top)\), denoting entities near to an airport that are not capitals, of which all positive nodes are instances and no negative nodes are instances. Such class descriptions are learnt in an analogous manner to how aforementioned systems like AMIE learn rules, with a refinement operator used to move from more general classes to more specific classes (and vice-versa), a confidence scoring function, and a search strategy. Another prominent such system is DL-Learner [Bühmann et al., 2016], which system further supports learning more general axioms through a scoring function that uses count queries to determine what ratio of expected edges – edges that would be entailed were the axiom true – are indeed found in the graph; for example, to score the axiom \(\exists\)flight\(^{-}\).DomesticAirport \(\sqsubseteq\) InternationalAirport over Figure 5.8, we can use a graph query to count how many nodes have incoming flights from a domestic airport (there are \(3\)), and how many nodes have incoming flights from a domestic airport and are international airports (there is \(1\)), where the greater the difference between both counts, the weaker the evidence for the axiom.

Hypothesis mining

We now provide some abstract formal definitions for the tasks of rule mining and axiom mining over graphs, which we generically refer to as hypothesis mining.

First we introduce hypothesis induction: a task that captures a more abstract (ideal) case for hypothesis mining. For simplicity, we focus on directed edge-labelled graphs. With a slight abuse of notation, we may interpret a set of edges \(E\) as the graph with precisely those edges and with no nodes or labels without edges. We may also interpret an edge \(e\) as the graph formed by \(\{ e \}\).

Hypothesis induction
The task of hypothesis induction assumes a particular graph entailment relation \(\models_\Phi\) (see Definition 4.4; hereafter simply \(\models\)). Given background knowledge in the form of a knowledge graph \(G\) (a directed edge-labelled graph, possibly extended with rules or ontologies), a set of positive edges \(E^{+}\) such that \(G\) does not entail any edge in \(E^{+}\) (i.e., for all \(e^{+} \in E^{+}\), \(G \not\models e^{+}\)) and \(E^{+}\) does not contradict \(G\) (i.e., there is a model of \(G \cup E^{+}\)), and a set of negative edges \(E^{-}\) such that \(G\) does not entail any edge in \(E^-\) (i.e., for all \(e^{-} \in E^{-}\), \(G \not\models e^{-}\)), the task is to find a set of hypotheses (i.e., a set of directed edge-labelled graphs) \(\Psi\) such that:
  • \(G \not\models \psi\) for all \(\psi \in \Psi\) (the background knowledge does not entail any hypothesis directly);
  • \(G \cup \Psi^* \models E^{+}\) (the background knowledge and hypotheses together entail all positive edges);
  • for all \(e^{-} \in E^{-}\), \(G \cup \Psi^* \not\models e^{-}\) (the background knowledge and hypotheses together do not entail any negative edge);
  • \(G \cup \Psi^* \cup E^{+}\) has a model (the background knowledge, hypotheses and positive edges taken together do not contain a contradiction);
  • for all \(e^{+} \in E^{+}\), \(\Psi^* \not\models e^{+}\) (the hypotheses alone do not entail a positive edge).
where by \(\Psi^* \coloneqq \cup_{\psi \in \Psi} \psi\) we denote the union of all graphs in \(\Psi\).

Let us assume ontological entailment \(\models\) with semantic conditions \(\Phi\) as defined in Tables 4.14.3. Given the graph of Figure 5.8 as the background knowledge \(G\), along with:

  • a set of positive edges \(E^{+} = \{ \)SCLflightARI, SCLdomestic flightARI\( \}\), and
  • a set of negative edges \(E^{-} = \{ \)ARIflightLIM, SCLdomestic flightLIM\( \}\),

then a set of hypotheses \(\Psi = \{ \)flighttypeSymmetric, domestic flighttypeSymmetric\( \}\) are not entailed by \(G\), entail all positive edges in \(E^{+}\) and no negative edges in \(E^{-}\) when combined with \(G\), do not contradict \(G \cup E^{+}\), and do not entail a positive edge without \(G\). Thus \(\Psi\) satisfies the conditions for hypothesis induction.

This task represents a somewhat idealised case. Often there is no set of positive edges distinct from the background knowledge itself. Furthermore, hypotheses not entailing a few positive edges, or entailing a few negative edges, may still be useful. The task of hypothesis mining rather accepts as input the background knowledge \(G\) and a set of negative edges \(E^{-}\) (such that for all \(e^{-} \in E^{-}\), \(G \not\models e^{-}\)), and attempts to score individual hypotheses \(\psi\) (such that \(G \not\models \psi\)) per their ability to “explain” \(G\) while minimising the number of elements of \(E^{-}\) entailed by \(G\) and \(\psi\). We can now abstractly define the task of hypothesis mining.

Hypothesis mining
Given a knowledge graph \(G\), a set of negative edges \(E^{-}\), a scoring function \(\sigma\), and a threshold \(\textsf{min}_{\sigma}\), the goal of hypothesis mining is to identify a set of hypotheses \(\{ \psi \mid G \not\models \psi\text{ and }\sigma(\psi,G,E^{-}) \geq \textsf{min}_{\sigma} \}\).

There are two scoring functions that are frequently used for \(\sigma\) in the literature: support and confidence.

Hypothesis support and confidence
Given a knowledge graph \(G = (V,E,L)\) and a hypothesis \(\psi\), the positive support of \(\psi\) is defined as: \[ \sigma^{+}(\psi,G) \coloneqq |\{ e \in E \mid G' \not\models e \text{ and }G' \cup \psi \models e \}| \] where \(G'\) denotes \(G\) with the edge \(e\) removed. Further given a set of negative edges \(E^{-}\), the negative support of \(\psi\) is defined as: \[ \sigma^{-}(\psi,G,E^{-}) \coloneqq |\{ e^{-} \in E^{-} \mid G \cup \psi \models e^{-} \}| \] Finally, the confidence of \(\psi\) is defined as \(\sigma^\pm(\psi,G,E^{-}) \coloneqq \frac{\sigma^{+}(\psi,G)}{\sigma^{+}(\psi,G) + \sigma^{-}(\psi,G,E^{-})}\).

We have yet to define how the set of negative edges are defined, which, in the context of a knowledge graph \(G\), depends on which assumption is applied:

  • Closed world assumption (CWA): For any (positive) edge \(e\), \(G \not\models e\) if and only if \(G \models \neg e\). Under CWA, any edge \(e\) not entailed by \(G\) can be considered a negative edge.
  • Open world assumption: For a (positive) edge \(e\), \(G \not\models e\) does not necessarily imply \(G \models \neg e\). Under OWA, the negation of an edge must be entailed by \(G\) for it to be considered negative.
  • Partial completeness assumption (PCA): If there exists an edge \((s,p,o)\) such that \(G \models (s,p,o)\), then for all \(o'\) such that \(G \not\models (s,p,o')\), it is assumed that \(G \models \neg(s,p,o')\). Under PCA, if \(G\) entails some outgoing edge(s) labelled \(p\) from a node \(s\), then such edges are assumed to be complete, and any edge \((s,p,o')\) not entailed by \(G\) can be considered a negative edge.

Knowledge graphs are generally incomplete – in fact, one of the main applications of hypothesis mining is to try to improve the completeness of the knowledge graph – and thus it would appear unwise to assume that any edge that is not currently entailed is false/negative. We can thus rule out CWA. Conversely, under OWA, potentially few (or no) negative edges might be entailed by the given ontologies/rules, and thus hypotheses may end up having low negative support despite entailing many edges that do not make sense in practice. Hence the PCA can be adopted as a heuristic to increase the number of negative edges and apply more sensible scoring of hypotheses. We remark that one can adapt PCA to define negative triples by changing the subject or predicate instead of the object.

Different implementations of hypothesis mining may consider different logical languages. Rule mining, for example, mines hypotheses expressed either as monotonic rules (with positive edges) or non-monotonic edges (possibly with negated edges). On the other hand, axiom mining considers hypotheses expressed in a logical language such as Description Logics. Particular implementations may, for practical reasons, impose further syntactic restrictions on the hypotheses generated, such as to impose thresholds on their length, on the symbols they use, or on other structural properties (such as “closed rules” in the case of the AMIE rule mining system [Galárraga et al., 2013]; see Section 5.4). Systems may further implement different search strategies for hypotheses. Systems such as DL-FOIL [Fanizzi et al., 2008, Rizzo et al., 2020], AMIE [Galárraga et al., 2013], RuLES [Ho et al., 2018], CARL [Pellissier Tanon et al., 2017], DL-Learner [Bühmann et al., 2016], etc., propose discrete mining that recursively generates candidate formulae through refinement/genetic operators that are then scored and checked for threshold criteria. On the other hand, systems such as NeuralLP [Yang et al., 2017] and DRUM [Sadeghian et al., 2019] apply differentiable mining that allows for learning (path-like) rules and their scores in a more continuous fashion (e.g., using gradient descent). We refer to Section 5.4 for further discussion and examples of such techniques for mining hypotheses.

Creation and Enrichment

In this chapter, we discuss the principal techniques by which knowledge graphs can be created and subsequently enriched from diverse sources of legacy data that may range from plain text to structured formats (and anything in between). The appropriate methodology to follow when creating a knowledge graph depends on the actors involved, the domain, the envisaged applications, the available data sources, etc. Generally speaking, however, the flexibility of knowledge graphs lends itself to starting with an initial core that can be incrementally enriched from other sources as required (typically following an Agile [Hunt and Thomas, 2003] or “pay-as-you-go” [Sequeda et al., 2019] methodology). For our running example, we assume that the tourism board decides to build a knowledge graph from scratch, aiming to initially describe the main tourist attractions – places, events, etc. – in Chile in order to help visiting tourists identify those that most interest them. The board decides to postpone adding further data, like transport routes, reports of crime, etc., for a later date.

Human Collaboration

One approach for creating and enriching knowledge graphs is to solicit direct contributions from human editors. Such editors may be found in-house (e.g., employees of the tourist board), using crowd-sourcing platforms, through feedback mechanisms (e.g., tourists adding comments on attractions), through collaborative-editing platforms (e.g., an attractions wiki open to public edits), etc. Though human involvement incurs high costs [Paulheim, 2018], some prominent knowledge graphs have been primarily based on direct contributions from human editors [Vrandečić and Krötzsch, 2014, He et al., 2016]. Depending on how the contributions are solicited, however, the approach has a number of key drawbacks, due primarily to human error [Pellissier Tanon et al., 2016], disagreement [Yasseri et al., 2012], bias [Janowicz et al., 2018], vandalism [Heindorf et al., 2016], etc. Successful collaborative creation further raises challenges concerning licensing, tooling, and culture [Pellissier Tanon et al., 2016]. Humans are sometimes rather employed to verify and curate additions to a knowledge graph extracted by other means [Pellissier Tanon et al., 2016] (through, e.g., video games with a purpose [Jurgens and Navigli, 2014]), to define high-quality mappings from other sources [Das et al., 2012], to define appropriate high-level schema [Keet, 2018, Labra Gayo et al., 2018], and so forth.

Text Sources

Text corpora – such as sourced from newspapers, books, scientific articles, social media, emails, web crawls, etc. – are an abundant source of rich information [Hellmann et al., 2013, Rospocher et al., 2016]. However, extracting such information with high precision and recall for the purposes of creating or enriching a knowledge graph is a non-trivial challenge. To address this, techniques from Natural Language Processing (NLP) [Maynard et al., 2016, Jurafsky and Martin, 2019] and Information Extraction (IE) [Weikum and Theobald, 2010, Grishman, 2012, Martínez-Rodríguez et al., 2020] can be applied. Though processes vary considerably across text extraction frameworks, in Figure 6.1 we illustrate four core tasks for text extraction on a sample sentence. We will discuss these tasks in turn.

Text extraction example; dashed nodes are new to the knowledge graph
Text extraction example; dashed nodes are new to the knowledge graph

Pre-processing

The pre-processing task may involve applying various techniques to the input text, where Figure 6.1 illustrates Tokenisation, which parses the text into atomic terms and symbols. Other pre-processing tasks applied to a text corpus may include: Part-of-Speech (POS) tagging [Maynard et al., 2016, Jurafsky and Martin, 2019] to identify terms representing verbs, nouns, adjectives, etc.; Dependency Parsing, which extracts a grammatical tree structure for a sentence where leaf nodes indicate individual words that together form phrases (e.g., noun phrases, verb phrases) and eventually clauses and sentences [Maynard et al., 2016, Jurafsky and Martin, 2019]; and Word Sense Disambiguation (WSD) [Navigli, 2009] to identify the meaning (aka sense) in which a word is used, linking words with a lexicon of senses (e.g., WordNet [Miller and Fellbaum, 2007] or BabelNet [Navigli and Ponzetto, 2012]), where, for instance, the term flights may be linked with the WordNet sense “an instance of travelling by air” rather than “a stairway between one floor and the next”. The appropriate type of pre-processing to apply often depends on the requirements of later tasks in the pipeline.

Named Entity Recognition (NER)

The NER task identifies mentions of named entities in a text [Nadeau and Sekine, 2007, Ratinov and Roth, 2009], typically targetting mentions of people, organisations, locations, and potentially other types [Ling and Weld, 2012, Nakashole et al., 2013, Yogatama et al., 2015]. A variety of NER techniques exist, with many modern approaches based on learning frameworks that leverage lexical features (e.g., POS tags, dependency parse trees, etc.) and gazetteers (e.g., lists of common first names, last names, countries, prominent businesses, etc.). Supervised methods [Bikel et al., 1999, Finkel et al., 2005, Lample et al., 2016] require manually labelling all entity mentions in a training corpus, whereas bootstrapping-based approaches [Collins and Singer, 1999, Etzioni et al., 2004, Nakashole et al., 2013, Gupta and Manning, 2014] rather require a small set of seed examples of entity mentions from which patterns can be learnt and applied to unlabelled text. Distant supervision [Ling and Weld, 2012, Ren et al., 2015, Yogatama et al., 2015] uses known entities in a knowledge graph as seed examples through which similar entities can be detected. Aside from learning-based frameworks, traditional approaches based on manually-crafted rules [Kluegl et al., 2009, Chiticariu et al., 2018] are still sometimes used due to their more controllable and predictable behaviour [Chiticariu et al., 2013]. The named entities identified by NER may be used to generate new candidate nodes for the knowledge graph (known as emerging entities, shown dashed in Figure 6.1), or may be linked to existing nodes per the Entity Linking task described in the following.

Entity Linking (EL)

The EL task associates mentions of entities in a text with the existing nodes of a target knowledge graph, which may be the nucleus of a knowledge graph under creation, or an external knowledge graph [Wu et al., 2018]. In Figure 6.1, we assume that the nodes Santiago and Easter Island already exist in the knowledge graph (possibly extracted from other sources). EL may then link the given mentions to these nodes. The EL task presents two main challenges. First, there may be multiple ways to mention the same entity, as in the case of Rapa Nui and Easter Island; if we created a node Rapa Nui to represent that mention, we would split the information available under both mentions across different nodes, where it is thus important for the target knowledge graph to capture the various aliases and multilingual labels by which one can refer to an entity [Moro et al., 2014]. Second, the same mention in different contexts can refer to distinct entities; for instance, Santiago can refer to cities in Chile, Cuba, Spain, amongst others. The EL task thus considers a disambiguation phase wherein mentions are associated to candidate nodes in the knowledge graph, the candidates are ranked, and the most likely node being mentioned is chosen [Wu et al., 2018]. Context can be used in this phase; for example, if Easter Island is a likely candidate for the corresponding mention alongside Santiago, we may boost the probability that this mention refers to the Chilean capital as both candidates are located in Chile. Other heuristics for disambiguation consider a prior probability, where for example, Santiago most often refers to the Chilean capital (being, e.g., the largest city with that name); centrality measures on the knowledge graph can be used for such purposes [Wu et al., 2018].

Relation Extraction (RE)

The RE task extracts relations between entities in the text [Zhou et al., 2005, Bach and Badaskar, 2007]. The simplest case is that of extracting binary relations in a closed setting wherein a fixed set of relation types are considered. While traditional approaches often relied on manually-crafted patterns [Hearst, 1992], modern approaches rather tend to use learning-based frameworks [Roller et al., 2018], including supervised methods over manually-labelled examples [Bunescu and Mooney, 2005, Zhou et al., 2005]. Other learning-based approaches again use bootstrapping [Etzioni et al., 2004, Bunescu and Mooney, 2007] and distant supervision [Mintz et al., 2009, Riedel et al., 2010, Hoffmann et al., 2011, Surdeanu et al., 2012, Xu et al., 2013, Smirnova and Cudré-Mauroux, 2019] to forgo the need for manual labelling; the former requires a subset of manually-labelled seed examples, while the latter finds sentences in a large corpus of text mentioning pairs of entities with a known relation/edge, which are used to learn patterns for that relation. Binary RE can also be applied using unsupervised methods in an open setting – often referred to as Open Information Extraction (OIE) [Banko et al., 2007, Etzioni et al., 2011, Fader et al., 2011, Mausam et al., 2012, Mausam, 2016, Mitchell et al., 2018] – whereby the set of target relations is not pre-defined but rather extracted from text based on, for example, dependency parse trees from which relations are taken.

A variety of RE methods have been proposed to extract \(n\)-ary relations that capture further context for how entities are related. In Figure 6.1, we see how an \(n\)-ary relation captures additional temporal context, denoting when Rapa Nui was named a World Heritage site; in this case, an anonymous node is created to represent the higher-arity relation in the directed-labelled graph. Various methods for \(n\)-ary RE are based on frame semantics [Fillmore, 1976], which, for a given verb (e.g., “named”), captures the entities involved and how they may be interrelated. Resources such as FrameNet [Baker et al., 1998] then define frames for words, which, for example, may identify that the semantic frame for “named” includes a speaker (the person naming something), an entity (the thing named) and a name. Optional frame elements are an explanation, a purpose, a place, a time, etc., that may add context to the relation. Other RE methods are rather based on Discourse Representation Theory (DRT) [Kamp, 1981], which considers a logical representation of text based on existential events. Under this theory, for example, the naming of Easter Island as a World Heritage Site is considered to be an (existential) event where Easter Island is the patient (the entity affected), leading to the logical (neo-Davidsonian) formula:

\( \exists e: \big(\)naming\((e),\) patient\((e,\) Easter Island\(),\) name\((e,\) World Heritage Site\()\big) \)

Such a formula is analogous to reification, as discussed previously in Section 3.3, where \(e\) is an existential term that refers to the \(n\)-ary relation being extracted.

Finally, while relations extracted in a closed setting are typically mapped directly to a knowledge graph, relations that are extracted in an open setting may need to be aligned with the knowledge graph; for example, if an OIE process extracts a binary relation Santiagohas flights toEaster Island, it may be the case that the knowledge graph does not have other edges labelled has flights to, where alignment may rather map such a relation to the edge SantiagoflightEaster Island assuming flight is used in the knowledge graph. A variety of methods have been applied for performing such alignments, including mappings [Corcoglioniti et al., 2016, Gangemi et al., 2017] and rules [Rouces et al., 2015] for aligning \(n\)-ary relations; distributional and dependency-based similarities [Moro and Navigli, 2013], association rule mining [Dutta et al., 2014], Markov clustering [Dutta et al., 2015] and linguistic techniques [Martínez-Rodríguez et al., 2018] for aligning OIE relations; amongst others.

Joint tasks

Having presented the four main tasks for building knowledge graphs from text, it is important to note that frameworks do not always follow this particular sequence of tasks. A common trend, for example, is to combine interdependent tasks, jointly performing WSD and EL [Moro et al., 2014], or NER and EL [Luo et al., 2015, Nguyen et al., 2016], or NER and RE [Ren et al., 2017, Zheng et al., 2017], etc., in order to mutually improve the performance of multiple tasks. For further details on extracting knowledge graphs from text we refer to the book by Maynard et al. [2016] and the recent survey by Martínez-Rodríguez et al. [2020].

Markup Sources

The Web was founded on interlinking markup documents wherein markers (aka tags) are used to separate elements of the document (typically for formatting purposes). Most documents on the Web use the HyperText Markup Language (HTML). Figure 6.2 presents an example HTML webpage about World Heritage Sites in Chile. Other formats of markup include Wikitext used by Wikipedia, TeX for typesetting, Markdown used by Content Management Systems, etc. One approach for extracting information from markup documents – in order to create and/or enrich a knowledge graph – is to strip the markers (e.g., HTML tags), leaving only plain text upon which the techniques from the previous section can be applied. However, markup can be useful for extraction purposes, where variations of the aforementioned tasks for text extraction have been adapted to exploit such markup [Lu et al., 2013, Lockard et al., 2018, Martínez-Rodríguez et al., 2020]. We can divide extraction techniques for markup documents into three main categories: general approaches that work independently of the markup used in a particular format, often based on wrappers that map elements of the document to the output; focussed approaches that target specific forms of markup in a document, most typically web tables (but sometimes also lists, links, etc.); and form-based approaches that extract the data underlying a webpage, per the notion of the Deep Web. These approaches can often benefit from the regularities shared by webpages of a given website; for example, intuitively speaking, while the webpage of Figure 6.2 is about Chile, we will likely find pages for other countries following the same structure on the same website.

<html>
  <head><title>UNESCO World Heritage Sites</title></head>
  <body>
    <h1>World Heritage Sites</h1>
	<h2>Chile</h2>
	<p>Chile has 6 UNESCO World Heritage Sites.</p>
	<table border="1">
	  <tr><th>Place</th><th>Year</th><th>Criteria</th></tr>
	  <tr><td>Rapa Nui</td><td>1995</td>
		<td rowspan="6">Cultural</td></tr>
	  <tr><td>Churches of Chiloé</td><td>2000</td></tr>
	  <tr><td>Historical Valparaíso</td><td>2003</td></tr>
	  <tr><td>Saltpeter Works</td><td>2005</td></tr>
	  <tr><td>Sewell Mining Town</td><td>2006</td></tr>
	  <tr><td>Qhapaq Ñan</td><td>2014</td></tr>
	</table>
  </body>
</html>
 

UNESCO World Heritage Sites

World Heritage Sites
Chile

Chile has 6 UNESCO World Heritage Sites.

PlaceYearCriteria
Rapa Nui1995 Cultural
Churches of Chiloé2000
Historical Valparaíso2003
Saltpeter Works2005
Sewell Mining Town2006
Qhapaq Ñan2014
 
Example markup document (HTML) with source-code (left) and formatted document (right)

Wrapper-based extraction

Many general approaches are based on wrappers that locate and extract the useful information directly from the markup document. While the traditional approach was to define such wrappers manually – a task for which a variety of declarative languages and tools have been defined – such approaches are brittle to changes in a website’s layout [Ferrara et al., 2014]. Hence other approaches allow for (semi-)automatically inducing wrappers [Flesca et al., 2004]. A modern such approach – used to enrich knowledge graphs in systems such as LODIE [Gentile et al., 2014] – is to apply distant supervision, whereby EL is used to identify and link entities in the webpage to nodes in the knowledge graph such that paths in the markup that connect pairs of nodes for known edges can be extracted, ranked, and applied to other examples. Taking Figure 6.2, for example, distant supervision may link Rapa Nui and World Heritage Sites to the nodes Easter Island and World Heritage Site in the knowledge graph using EL, and given the edge Easter IslandnamedWorld Heritage Site in the knowledge graph (extracted per Figure 6.1), identify the candidate path \((x,\)td\([1]^{-} \cdot \) tr\(^{-} \cdot \) table\(^- \cdot \) h1\(,y)\) as reflecting edges of the form \(x\)named\(y\), where \(t[n]\) indicates the \(n\)th child of tag \(t\), \(t^-\) its inverse, and \(t_1 \cdot t_2\) concatenation. Finally, paths with high confidence (e.g., ones “witnessed” by many known edges in the knowledge graph) can then be used to extract novel edges, such as Qhapaq ÑannamedWorld Heritage Site, both on this page and on related pages of the website with similar structure (e.g., for other countries).

Web table extraction

Other approaches target specific types of markup, most commonly web tables embedded in HTML webpages. However, web tables are designed to enhance human rather than machine readability. Many web tables are used for layout and page structure (e.g., navigation bars). Those that contain data may follow different formats, such as relational tables, listings, attribute-value tables, and matrices [Cafarella et al., 2008, Crestan and Pantel, 2011]. A first step is to classify tables to find ones appropriate for the given extraction mechanism(s) [Crestan and Pantel, 2011, Eberius et al., 2015]. Next, web tables may contain column spans, row spans, inner tables, or may be split vertically to improve human aesthetics. Table normalisation merges split tables, un-nests tables, transposes tables, etc. [Pivk et al., 2007, Cafarella et al., 2008, Crestan and Pantel, 2011, Deng et al., 2013, Ermilov and Ngonga Ngomo, 2016, Lehmberg et al., 2016]. Some approaches then identify the table protagonist [Crestan and Pantel, 2011, Muñoz et al., 2014] – the main entity that the table describes – often found elsewhere in the webpages; for example, though not mentioned by the table of Figure 6.1, World Heritage Sites is its protagonist. Finally, extraction processes may associate cells with entities [Limaye et al., 2010, Mulwad et al., 2013], columns with types [Deng et al., 2013, Limaye et al., 2010, Mulwad et al., 2013], and column pairs with relations [Limaye et al., 2010, Muñoz et al., 2014]. When enriching knowledge graphs, recent approaches apply distant supervision, linking cells to knowledge graph nodes in order to generate candidates for type and relation extraction [Limaye et al., 2010, Mulwad et al., 2013, Muñoz et al., 2014]. Statistical distributions can also help to link numerical columns [Neumaier et al., 2016]. Specialised table extraction frameworks have also been proposed for specific websites, where prominent knowledge graphs, such as DBpedia [Lehmann et al., 2015] and YAGO [Suchanek et al., 2008] focus on extraction from info-box tables in Wikipedia.

Deep Web crawling

The Deep Web presents a rich source of information accessible only through searches on web forms, thus requiring Deep Web crawling techniques to access [Madhavan et al., 2008]. Systems have been proposed to extract knowledge graphs from Deep Web sources [Geller et al., 2008, Lehmann et al., 2012, Collarana et al., 2016]. Approaches typically attempt to generate sensible form inputs – which may be based on a user query or generated from reference knowledge – and then extract data from the generated responses (markup documents) using the aforementioned techniques [Geller et al., 2008, Lehmann et al., 2012, Collarana et al., 2016].

Structured Sources

Much of the legacy data available within organisations and on the Web is represented in structured formats, primarily tables – in the form of relational databases, CSV files, etc. – but also tree-structured formats such as JSON, XML etc. Unlike text and markup documents, structured sources can often be mapped to knowledge graphs whereby the structure is (precisely) transformed according to a mapping rather than (imprecisely) extracted. The mapping process involves two steps: 1) create a mapping from the source to a graph, and 2) use the mapping in order to materialise the source data as a graph or to virtualise the source (creating a graph view over the legacy data).

Mapping from tables

Tabular sources of data are prevalent; for example, the structured content underlying many organisations and websites are housed in relational databases. In Figure 6.3 we present an example of a relational database instance that we wish to integrate into our knowledge graph. There are then two approaches for mapping content from tables to knowledge graphs: a direct mapping, and a custom mapping.

Report

crime claimant station date
Pickpocketing XY12SDA Viña del Mar 2019-04-12
Assault AB9123N Arica 2019-04-12
Pickpocketing XY12SDA Rapa Nui 2019-04-12
Fraud FI92HAS Arica 2019-04-13
 

Claimant

id name country
XY12SDA John Smith U.S.
AB9123N Joan Dubois France
XI92HAS Jorge Hernández Chile
 
Relational database instance with two tables describing crime data
Direct mapping result for the first rows of both tables in Figure 33
Direct mapping result for the first rows of both tables in Figure 6.3

A direct mapping automatically generates a graph from a table. We present in Figure 6.4 the result of a standard direct mapping [Arenas et al., 2012], which creates an edge xyz for each (non-header, non-empty, non-null) cell of the table, such that x represents the row of the cell, y the column name of the cell, and z the value of the cell. In particular, x typically encodes the values of the primary key for a row (e.g., Claimant.id); otherwise, if no primary key is defined (e.g., per the Report table), x can be an anonymous node or a node based on the row number. The node x and edge label y further encode the name of the table to avoid clashes across tables that have the same column names used with different meanings. For each row x, we may add a type edge based on the name of its table. The value z may be mapped to datatype values in the corresponding graph model based on the source domain (e.g., a value in an SQL column of type Date can be mapped to xsd:date in the RDF data model). If the value is null (or empty), typically the corresponding edge will be omitted.30note 30 One might consider representing nulls with anonymous/blank nodes. However, nulls in SQL can be used to mean that there is no such value, which conflicts with the existential semantics of such nodes (e.g., in RDF). With respect to Figure 6.4, we highlight the difference between the nodes Claimant-XY12SDA and XY12SDA, where the former denotes the row (or entity) identified by the latter primary key value. In case of a foreign key between two tables – such as Report.claimant referencing Claimant.id – we can link, for example, to Claimant-XY12SDA rather than XY12SDA, where the former node also has the name and country of the claimant. A direct mapping along these lines has been standardised for mapping relational databases to RDF [Arenas et al., 2012], where Stoica et al. [2019] have recently proposed an analogous direct mapping for property graphs. Another direct mapping has been defined for CSV and other tabular data [Tandy et al., 2015] that further allows for specifying column names, primary/foreign keys, and data types – which are often missing in such data formats – as part of the mapping itself.

Although a direct mapping can be applied automatically on tabular sources of data and preserve the information of the original source – i.e., allowing a deterministic inverse mapping that reconstructs the tabular source from the output graph [Sequeda et al., 2012] – in many cases it is desirable to customise a mapping, such as to align edge labels or nodes with a knowledge graph under enrichment, etc. Along these lines, declarative mapping languages allow for manually defining custom mappings from tabular sources to graphs. A standard language along these lines is the RDB2RDF Mapping Language (R2RML) [Das et al., 2012], which allows for mapping from individual rows of a table to one or more custom edges, with nodes and edges defined either as constants, as individual cell values, or using templates that concatenate multiple cell values from a row and static substrings into a single term; for example, a template {id}-{country} may produce nodes such as XY12SDA-U.S. from the Claimant table. In case that the desired output edges cannot be defined from a single row, R2RML allows for (SQL) queries to generate tables from which edges can be extracted where, for example, edges such as U.S.crimes2 can be generated by defining the mapping with respect to a query that joins the Report and Claimant tables on claimant=id, grouping by country, and applying a count for each country group. A mapping can then be defined on the results table such that the source node denotes the value of country, the edge label is the constant crimes, and the target node is the count value. An analogous standard also exists for mapping CSV and other tabular data to RDF graphs, again allowing keys, column names, and datatypes to be chosen as part of the mapping [Tennison and Kellogg, 2015].

Once the mappings have been defined, one option is to use them to materialise graph data following an Extract-Transform-Load (ETL) approach, whereby the tabular data are transformed and explicitly serialised as graph data using the mapping. A second option is to use virtualisation through a Query Rewriting (QR) approach, whereby queries on the graph (using, e.g., SPARQL, Cypher, etc.) are translated to queries over the tabular data (typically using SQL). Comparing these two options, ETL allows the graph data to be used as if they were any other data in the knowledge graph. However, ETL requires updates to the underlying tabular data to be explicitly propagated to the knowledge graph, whereas a QR approach only maintains one copy of data to be updated. The area of Ontology-Based Data Access (OBDA) [Xiao et al., 2018] is concerned with QR approaches that support ontological entailments as seen in Chapter 4. Although most QR approaches only support non-recursive entailments expressible as a single (non-recursive) query, some QR approaches support recursive entailments through rewritings to recursive queries [Sequeda et al., 2014].

Mapping from trees

A number of popular data formats are based on trees, including XML and JSON. While one could imagine – leaving aside issues such as the ordering of children in a tree – a trivial direct mapping from trees to graphs by simply creating edges of the form \(x\)child\(y\) for each node \(y\) that is a child of \(x\) in the source tree, such an approach is not typically used, as it represents the literal structure of the source data. Instead, the content of tree-structured data can be more naturally represented as a graph using a custom mapping. Along these lines, the GRDLL standard [Connolly, 2007] allows for mapping from XML to (RDF) graphs, while languages such as RML allow for mapping from a variety of formats, including XML and JSON, to (RDF) graphs [Dimou et al., 2014]. In contrast, hybrid query languages such as XSPARQL [Bischof et al., 2012] allow for querying XML and RDF in unison, thus supporting both materialisation and virtualisation of graphs over tree-structured sources of legacy data.

Mapping from other knowledge graphs

We may also leverage existing knowledge graphs in order to construct or enrich another knowledge graph. For example, a large number of points of interest for the Chilean tourist board may be available in existing knowledge graphs such as BabelNet [Navigli and Ponzetto, 2012], DBpedia [Lehmann et al., 2015], LinkedGeoData [Stadler et al., 2012], Wikidata [Vrandečić and Krötzsch, 2014], YAGO [Hoffart et al., 2011], etc. However, not all entities and/or relations may be of interest. A standard option to extract a relevant sub-graph of data is to use construct queries that generate graphs as output [Neumaier and Polleres, 2019]. Entity and schema alignment between the knowledge graphs may be further necessary to better integrate (parts of) external knowledge graphs, using linking tools for graphsexternal identifiers [Pellissier Tanon et al., 2016], or indeed may be done manually [Pellissier Tanon et al., 2016]. For instance, Wikidata [Vrandečić and Krötzsch, 2014] uses Freebase [Bollacker et al., 2007b, Pellissier Tanon et al., 2016] as a source; Gottschalk and Demidova [2018] extract an event-centric knowledge graph from Wikidata, DBpedia and YAGO; while Neumaier and Polleres [2019] construct a spatio-temporal knowledge graph from Geonames, Wikidata, and PeriodO [Golden and Shaw, 2016] (as well as tabular data).

Schema/Ontology Creation

The discussion thus far has focussed on extracting data from external sources in order to create and enrich a knowledge graph. In this section, we discuss some of the principal methods for generating a schema based on external sources of data, including human knowledge. For discussion on extracting a schema from the knowledge graph itself, we refer back to Section 3.1.3. In general, much of the work in this area has focussed on the creation of ontologies using either ontology engineering methodologies, and/or ontology learning. We discuss these two approaches in turn.

Ontology engineering

Ontology engineering refers to the development and application of methodologies for building ontologies, proposing principled processes by which better quality ontologies can be constructed and maintained with less effort. Early methodologies [Grüninger and Fox, 1995a, Fernández et al., 1997, Noy and McGuinness, 2001] were often based on a waterfall-like process, where requirements and conceptualisation were fixed before starting to define the ontology, using, for example, an ontology engineering tool [Gómez-Pérez et al., 2006, Keet, 2018, Kendall and McGuinness, 2019]. However, for situations involving large or ever-evolving ontologies, more iterative and agile ways of building and maintaining ontologies have been proposed.

DILIGENT [Pinto et al., 2009] was an early example of an agile methodology, proposing a complete process for ontology life-cycle management and knowledge evolution, as well as separating local changes (local views on knowledge) from global updates of the core part of the ontology, using a review process to authorise the propagation of changes from the local to the global level. This methodology is similar to how, for instance, the large clinical reference terminology SNOMED CT [IHTSDO, 2019] (also available as an ontology) is maintained and evolved, where the (international) core terminology is maintained based on global requirements, while national or local extensions to SNOMED CT are maintained based on local requirements. A group of authors then decides which national or local extensions to propagate to the core terminology. More modern agile methodologies include eXtreme Design (XD) [Presutti et al., 2009, Blomqvist et al., 2016], Modular Ontology Modelling (MOM) [Krisnadhi and Hitzler, 2016b, Hitzler and Krisnadhi, 2018], Simplified Agile Methodology for Ontology Development (SAMOD) [Peroni, 2016], and more besides. Such methodologies typically include two key elements: ontology requirements and (more recently) ontology design patterns.

Ontology requirements specify the intended task of the resulting ontology, or of the knowledge graph itself in conjunction with the new ontology. A common way to express ontology requirements is through Competency Questions (CQ) [Grüninger and Fox, 1995b], which are natural language questions illustrating the typical information needs that one would require the ontology (or the knowledge graph) to respond to. Such CQs can then be complemented with additional restrictions, and reasoning requirements, in case that the ontology should also contain restrictions and general axioms for inferring new knowledge or checking data consistency. A common way of testing ontologies (or knowledge graphs based on them) is then to formalise the CQs as queries over some test set of data, and make sure the expected results are entailed [Blomqvist et al., 2012, Keet and Ławrynowicz, 2016]. We may, for example, consider the CQ “What are all the events happening in Santiago?”, which can be represented as a graph query Eventtype?eventlocationSantiago. Taking the data graph of Figure 2.1 and the axioms of Figure 3.2, we can check to see if the expected result EID15 is entailed by the ontology and the data, and since it is not, we may consider expanding the axioms to assert that locationtypeTransitive.

Ontology Design Patterns (ODPs) are another common feature of modern methodologies [Gangemi, 2005, Blomqvist and Sandkuhl, 2005], specifying generalisable ontology modelling patterns that can be used as inspiration for modelling similar patterns, as modelling templates [Egaña et al., 2008, Skjæveland et al., 2018], or as directly reusable components [Daga et al., 2008, Shimizu et al., 2019]. Several pattern libraries have been made available online, ranging from carefully curated ones [Aranguren et al., 2008, Shimizu et al., 2019] to open and community moderated ones [Daga et al., 2008]. As an example, to model events in our scenario, we may adopt the Core Event ontology pattern proposed by Krisnadhi and Hitzler [2016a], which specifies a spatio-temporal extent, sub-events, and participants of an event, along with competency questions, formal definitions, etc., to support this pattern.

Ontology learning

The previous methodologies outline methods by which ontologies can be built and maintained manually. Ontology learning, in contrast, can be used to (semi-)automatically extract information from text that is useful for the ontology engineering process [Buitelaar et al., 2005, Cimiano, 2006]. Early methods focussed on extracting terminology from text that may represent the relevant domain’s classes; for example, from a collection of text documents about tourism, a terminology extraction tool – using measures of unithood that determine how cohesive an \(n\)-gram is as a unitary phrase, and termhood that determine how relevant the phrase is to a domain [Martínez-Rodríguez et al., 2018] – may identify \(n\)-grams such as “visitor visa”, “World Heritage Site”, “off-peak rate”, etc., as terminology of particular importance to the tourist domain that thus may merit inclusion in such an ontology. Ontological axioms may also be extracted from text. A common target is to extract sub-class axioms from text, leveraging patterns based on modifying nouns and adjectives that incrementally specialise concepts (e.g., extracting Visitor Visasubc. ofVisa from the noun phrase “visitor visa” and isolated appearances of “visa” elsewhere), or using Hearst patterns [Hearst, 1992] (e.g., extracting Off-Peak Ratesubc. ofDiscount from “many discounts, such as off-peak rates, are available” based on the pattern “X, such as Y”). Textual definitions can also be harvested from large texts to extract hypernym relations and induce a taxonomy from scratch [Velardi et al., 2013]. More recent works aim to extract more expressive axioms from text, including disjointness axioms [Völker et al., 2015]; and axioms involving the union and intersection of classes, along with existential, universal, and qualified-cardinality restrictions [Petrucci et al., 2016]. The results of an ontology learning process can then serve as input to a more general ontology engineering methodology, allowing us to validate the terminological coverage of an ontology, to identify new classes and axioms, etc.

Quality Assessment

Independently of the (kinds of) source(s) from which a knowledge graph is created, the resulting initial knowledge graph will usually be incomplete, and will often contain duplicate, contradictory or even incorrect statements, especially when taken from multiple sources. After the initial creation and enrichment of a knowledge graph from external sources, a crucial step is thus to assess the quality of the resulting knowledge graph. By quality, we here refer to fitness for purpose. Quality assessment then helps to ascertain for which purposes a knowledge graph can be reliably used. Take, for instance, the sample of an initial knowledge graph created by the tourist board shown in Figure 7.1. Is this knowledge graph of good quality? Does it exhibit issues that might limit the applications for which it is fit for purpose? Can we define and detect such issues? These questions are crucial to address before the knowledge graph is deployed, but they are also challenging to address in a general way.

A newly created knowledge graph about events and their venues
A newly created knowledge graph about events and their venues

This chapter discusses (sometimes overlapping) quality dimensions that capture qualitative aspects of the multifaceted notion of data quality; some of these dimensions apply more generally to databases [Batini et al., 2015], while others are more specific to knowledge graphs [Zaveri et al., 2016]. We further discuss quality metrics that provide ways to measure quantitative aspects of these dimensions. We group dimensions and metrics in a manner inspired by Batini and Scannapieco [2016].

Accuracy

Accuracy refers to the extent to which entities and relations – encoded by nodes and edges in the graph – correctly represent real-life phenomena. Accuracy can be divided into three dimensions: syntactic accuracy, semantic accuracy, and timeliness.

Syntactic accuracy

Syntactic accuracy is the degree to which the data are accurate with respect to the grammatical rules defined for the domain and/or data model. A prevalent example of syntactic inaccuracy occurs with datatype nodes, which may be incompatible with a defined range or be malformed. For example, assuming that a property start is defined with the range xsd:dateTime, the value March 29, 2019 in Figure 7.1 would be incompatible with the defined range, while a value “March 29, 2019, 20:00”^^xsd:dateTime would be malformed (a value such as “2019-03-22T20:00:00”^^xsd:dateTime is rather expected). A corresponding metric for syntactic accuracy is the ratio between the number of invalid values of a given property and the total number of values for the same property [Zaveri et al., 2016]. Such forms of syntactic accuracy can typically be assessed using validation tools [Fürber and Hepp, 2011, Hogan et al., 2010].

Semantic accuracy

Semantic accuracy is the degree to which data values correctly represent real-world phenomena, which may be affected by imprecise extraction results, untrustworthy sources, vandalism, etc. For instance, in Figure 7.1, the start of the EID15 event comes after the end of the event, possibly due to a typo in the year. While such a case could potentially be identified using, for example, shape-based validation, other cases might be more difficult to detect; for example, if we were to accidentally (and incorrectly) swap the venues for EID15 and EID17, there might be no indication whatsoever in the knowledge graph that the venues are incorrect, even if we have additional schemata/ontologies/rules available. Assessing the level of semantic inaccuracy is challenging. While one option is to apply manual verification, an automatic option may be to check the stated relation against several sources [Lei et al., 2007, Esteves et al., 2018]. An alternative is to validate the quality of the processes used to generate the knowledge graph, based on measures such as precision, possibly with the help of human experts or gold standards [Martínez-Rodríguez et al., 2020].

Timeliness

Timeliness is the degree to which the knowledge graph is kept up-to-date with the real world state [Käfer et al., 2013]. A knowledge graph may be semantically accurate now, but may quickly become inaccurate (outdated) if no procedures are in place to keep it up-to-date in a timely manner. Considering Figure 7.1, the events appear to be from years ago, and if not updated, then the knowledge graph will not be suitable for applications that wish to recommend upcoming events to users. Additionally, the meaning of some values in the graph, such as Next Tuesday or Next Thursday (which may have been extracted from the text of a news article, for example), will change over time, and become semantically inaccurate in the future. Similarly, the age of Santiago will quickly become outdated, where instead representing the year that the city was founded would facilitate timeliness. Timeliness can be assessed based on how frequently the knowledge graph is updated with respect to underlying sources [Käfer et al., 2013, Rula et al., 2014], which can be done using temporal annotations of changes in the knowledge graph [Rula et al., 2012, Rula et al., 2019], as well as contextual representations that capture the temporal validity of data (see Section 3.3).

Coverage

Coverage refers to avoiding the omission of domain-relevant elements, which otherwise may yield incomplete query results or entailments, biased models, etc.

Completeness

Completeness refers to the degree to which all required information is present in a particular dataset. Completeness comprises the following aspects: (i) schema completeness refers to the degree to which the classes and properties of a schema are represented in the data graph, (ii) property completeness refers to the ratio of missing values for a specific property, (iii) population completeness refers to the percentage of all real-world entities of a particular type that are represented in the datasets, and (iv) linkability completeness refers to the degree to which instances in the data set are interlinked. Taking some examples from Figure 7.1, the lack of information about the fare for EID15 might be seen as a form of property incompleteness, while missing events held in Chile around the same time might lead to population incompleteness. Measuring completeness is non-trivial as it assumes knowledge of a hypothetical ideal knowledge graph [Darari et al., 2018] that contains all the elements that the knowledge graph in question should have. Concrete strategies may involve comparison with gold standards that provide samples of the ideal knowledge graph (possibly based on completeness statements [Darari et al., 2018]), or measuring the recall of extraction methods from complete sources [Martínez-Rodríguez et al., 2020].

Representativeness

Representativeness is a related dimension that, instead of focusing on the ratio of domain-relevant elements that are missing, rather focuses on assessing high-level biases in what is included/excluded from the knowledge graph [Baeza-Yates, 2018]. As such, this dimension assumes that the knowledge graph is incomplete – i.e., that it is a sample of the ideal knowledge graph – and asks how biased this sample is. Biases may occur in the data, in the schema, or during reasoning [Janowicz et al., 2018]. Examples of data biases include geographic biases that under-represent entities/relations from certain parts of the world [Janowicz et al., 2018], linguistic biases that under-represent multilingual resources (e.g., labels and descriptions) for certain languages [Kaffee et al., 2017], social biases that under-represent people of particular genders or races [Wagner et al., 2016], and so forth. In contrast, schema biases may result from high-level definitions extracted from biased data [Janowicz et al., 2018], semantic definitions that do not cover uncommon cases, etc. Unrecognised biases may lead to adverse effects; for example, if the knowledge graph of Figure 7.1 has a geographic bias towards events and attractions close to Santiago city – due perhaps to the sources used for creation, the employment of curators from the city, etc. – then this may lead to tourism in and around Santiago being disproportionately promoted to the detriment of tourism elsewhere in Chile. Measures of representativeness may involve comparing known statistical distributions with those of the knowledge graph, for example, comparing geolocated entities with known population densities [Janowicz et al., 2018], linguistic distributions with known distributions of speakers [Kaffee et al., 2017], etc. Another more general option is to compare the knowledge graph with general statistical laws, where Soulet et al. [2018] use (non-)conformance with Benford’s law31note 31 Benford’s law states that the leading significant digit in many collections of numbers is more likely to be small. to measure representativeness in knowledge graphs.

Coherency

Coherency refers to how well the knowledge graph conforms to – or is coherent with – the formal semantics and constraints defined at the schema-level.

An ontology for the knowledge graph of Figure 7.1
An ontology for the knowledge graph of Figure 7.1

Consistency

Consistency means that a knowledge graph is free of contradictions (i.e., inconsistencies) with respect to the particular logical entailment considered. For example, if we apply the entailments defined in Table 4.1 over the graph of Figure 7.1, we see that the edge Santiago de Chilesame asSantiago de Cuba is inferred from both entities being the same as Santiago, which generates an inconsistency with the edge Santiago de Chilediff. fromSantiago de Cuba as stated in the graph. While in this case it is evident that Santiago de Cubasame asSantiago is semantically inaccurate (considering that the venues connected to Santiago are in Chile), in other cases there may not be an obvious inaccuracy. Take, for example, the ontology defined in Figure 7.2, combined with the graph of Figure 7.1, and the ontological entailments of Tables 4.14.3. Noting that the food festival EID15 offers a takeaway service, according to the ontology, this entails that EID15 is a restaurant, a building, and a place, which is disjoint with event. However, EID15 is also entailed to be a festival, and then an event, generating an inconsistency. In this case there is no clear individual “error” leading to an inconsistency. Possibly the graph of Figure 7.1 should not use the property service for a food event (though it seems a “good fit”), or perhaps the ontology of Figure 7.2 should not define the domain of the property service to be a restaurant. Any ontological features in Tables 4.14.3 with a “not” condition can give rise to inconsistencies if the negated condition is entailed. A measure of consistency can be the number of inconsistencies found in a knowledge graph, possibly sub-divided into the number of such inconsistencies identified by each semantic feature [Bonatti et al., 2011].

Validity

Validity means that the knowledge graph is free of constraint violations, such as captured by shape expressions [Thornton et al., 2019] (see Section 3.1.2. We may, for example, specify a shape City whose target nodes have at most one country. Then, taking the edges ChilecountrySantiagocountryCuba from Figure 7.1, and assuming that Santiago becomes a target of City, we have a constraint violation. Conversely, even if we defined analogous cardinality restrictions in an ontology (e.g., even if we defined that country was functional), this would not necessarily cause an inconsistency since, without UNA, we would first infer that Chile and Cuba refer to the same entity. Similarly, using shapes, we can more easily detect missing data; for example, we can define a shape Event, and require that it have at least one value for the property fare. Now, if EID15 becomes targetted by Event, then we will have a constraint violation as the node has no value for fare. Conversely, even if we defined analogous cardinality restrictions in an ontology (e.g., we defined that events have a minimum cardinality of 1 for fare), this would not cause an inconsistency since, under the OWA, we would rather entail that the event EID15 has some fair (that is not described in the graph). Consistency and validity can thus indicate different types of issues. A straightforward measure of validity is to count the number of violations per constraint.

Succinctness

Succinctness refers to the inclusion only of relevant content (avoiding “information overload”) that is represented in a concise and intelligible manner.

Conciseness

Conciseness refers to avoiding schema and data elements that are irrelevant to the domain. Mendes et al. [2012b] distinguish intensional conciseness (schema level), which refers to the case when the data do not contain redundant schema elements (properties, classes, shapes, etc.), and extensional conciseness (data level), where the data do not describe redundant entities and relations. For example, the inclusion of a property and class for modelling jurisdictions and legal entities in the ontology of Figure 7.2 may affect the intensional conciseness of the ontology in the context of a knowledge graph about tourist events. Similarly, the inclusion of data about Santiago de Cuba in our knowledge graph dedicated to tourism in Chile may affect the extensional conciseness of the knowledge graph, potentially returning irrelevant results for the given domain. In general, conciseness can be measured in terms of the ratio of properties, classes, shapes, entities, relations, etc., of relevance to the domain, which may in turn require a gold standard, or measures of domain-relevance.

Representational conciseness

Representational conciseness refers to the extent to which content is compactly represented in the knowledge graph, which may again be intensional or extensional [Zaveri et al., 2016]. For example, having two properties category and type serving the same purpose would negatively affect the intensional form of representational conciseness, while having two nodes Santiago and Santiago de Chile that split the data available about the capital of Chile would affect the extensional form of representational conciseness. Another example of poor representational conciseness is the unnecessary use of complex modelling constructs, such as using reification unnecessarily, or using linked lists when the order of elements is not important [Hogan et al., 2012a]. An example of this is the anonymous node used in Figure 7.1 to represent the days on which EID17 starts and ends, which could rather be directly associated with the event (at least if we assume that events have one start and one end moment in time). A different example is the specification of the duration of EID15, which could be calculated from the start and end values (assuming the correct datatypes were used). Though representational conciseness is challenging to assess, measures such as the number of redundant nodes can be used [Fürber and Hepp, 2011].

Understandability

Understandability refers to the ease with which data can be interpreted without ambiguity by human users, which involves – at least – the provision of human-readable labels and descriptions (preferably in different languages [Kaffee et al., 2017]) that allow such beings to understand what is being spoken about [Hogan et al., 2012a]. Referring back to Figure 7.1, though the nodes EID15 and EID17 are used to ensure unique identifiers for events, they should also be associated with labels, such as Ñam. Ideally the human readable information is sufficient to disambiguate a particular node, such as associating a description "Santiago, the capital of Chile"@en with Santiago to disambiguate the city from synonymous ones. Measures of understandability may include the ratio of nodes with human-readable labels and descriptions, the uniqueness of such labels and descriptions, the languages supported, etc.

Other Quality Dimensions

The list of quality dimensions provided here should be considered illustrative rather than complete. Further dimensions may be pertinent in the context of specific domains, applications, or graph data models. For more discussion, we refer to the survey by Zaveri et al. [2016] and to the book by Batini and Scannapieco [2016].

Refinement

Beyond assessing the quality of a knowledge graph, there exist techniques to refine the knowledge graph, in particular to (semi-)automatically complete and correct the knowledge graph [Paulheim, 2017], aka knowledge graph completion and knowledge graph correction, respectively. As distinguished from the creation and enrichment tasks outlined in Chapter 6, refinement typically does not involve applying extraction or mappings over external sources in order to ingest their content into a given knowledge graph (potentially using external sources to verify its content).

Completion

Knowledge graphs are characterised by incompleteness [West et al., 2014]. As such, knowledge graph completion aims at filling in the missing edges (aka missing links) of a knowledge graph, i.e., edges that are deemed correct but are neither given nor entailed by the knowledge graph. This task is often addressed with link prediction techniques proposed in the area of Statistical Relational Learning [Getoor and Taskar, 2007], which predict the existence – or sometimes more generally, predict the probability of correctness – of missing edges. For instance, one might predict that the edge Moon ValleybusSan Pedro is a probable missing edge for the graph of Figure 5.2, given that most bus routes observed are return services (i.e., bus is typically symmetric). Link prediction may target three settings: general links involving edges with arbitrary labels, e.g., bus, flight, type, etc.; type links involving edges with label type, indicating the type of an entity; and identity links involving edges with label same as, indicating that two nodes refer to the same entity (cf. Section 3.2.2). While type and identity links can be addressed using general link prediction techniques, the particular semantics of type and identity links can be addressed with custom techniques. (The related task of generating links across knowledge graphs – referred to as link discovery [Nentwig et al., 2017] – will be discussed later in Section 9.1.)

Link prediction, in the general case, is often addressed with inductive techniques as discussed in Chapter 5, and in particular, knowledge graph embeddings and rule/axiom mining. For example, given Figure 5.2, using knowledge graph embeddings, we may detect that given an edge of the form \(x\)bus\(y\), a (missing) edge \(y\)bus\(x\) has high plausibility, while using symbol-based approaches, we may learn the high-level rule ?xbus?y \(\Rightarrow\) ?ybus?x that may infer/predict new bus links. Either approach would help us to predict the missing link Moon ValleybusSan Pedro.

Type links are of particular importance to a knowledge graph, where dedicated techniques can be leveraged taking into account the specific semantics of such links. In the case of type prediction, there is only one edge label (type) and typically fewer distinct values (classes) than in other cases, such that the task can be reduced to a traditional classification task [Paulheim, 2017], training models to identify each semantic class based on features such as outgoing and/or incoming edge labels on their instances in the knowledge graph [Paulheim and Bizer, 2013, Sleeman and Finin, 2013]. For example, assume that in Figure 5.2 we also know that Arica, Calama, Puerto Montt, Punta Arenas and Santiago are of type City. We may then predict that Iquique and Easter Island are also of type City based on the presence of edges labelled flight to/from these nodes, which (we assume) are learnt to be a good feature for prediction of that class (the former prediction is correct, while the latter is incorrect). Graph neural networks (see Section 5.3) can also be used for node classification/type prediction.

Predicting identity links involves searching for nodes that refer to the same entity, but are not stated or entailed to be the same; this is analogous to the task of entity matching (aka record linkage, deduplication, etc.) considered in more general data integration settings [Köpcke and Rahm, 2010]. Such techniques are generally based on two types of matchers: value matchers determine how similar the values of two entities on a given property are, which may involve similarity metrics on strings, numbers, dates, etc.; while context matchers consider the similarity of entities based on various nodes and edges [Köpcke and Rahm, 2010]. An illustrative example is given in Figure 8.1, where value matchers will compute similarity between values such as 7400 and 7500, while context matchers will compute similarity between Easter Island and Rapa Nui based on their surrounding information, such as similar latitudes, longitudes, populations, and the same seat (conversely, a value matcher on this pair of nodes would measure string similarity between “Easter Island” and “Rapa Ñui”).

Identity linking example: Easter Island and Rapa Nui denote the same place
Identity linking example: Easter Island and Rapa Nui denote the same place

A major challenge in this setting is efficiency, where a pairwise matching would require \(O(n^2)\) comparisons for \(n\) the number of nodes. To address this issue, blocking can be used to group similar entities into (possibly overlapping, possibly disjoint) “blocks” based on similarity-preserving keys, with matching performed within each block [Isele et al., 2011, Köpcke and Rahm, 2010, Draisbach and Naumann, 2011]; for example, if matching places based on latitude/longitude, blocks may represent geographic regions. An alternative to discrete blocking is to use windowing over entities in a similarity-preserving ordering [Draisbach and Naumann, 2011], or to consider searching for similar entities within multi-dimensional spaces (e.g., spacetime [Santipantakis et al., 2019], spaces with Minkowski distances [Ngonga Ngomo, 2012], orthodromic spaces [Ngonga Ngomo, 2013], etc. [Sherif and Ngonga Ngomo, 2018]). The results can either be pairs of nodes with a computed confidence of them referring to the same entity, or crisp identity links extracted based on a fixed threshold, or binary classification [Köpcke and Rahm, 2010]. For confident identity links, the nodes’ edges may then be consolidated [Hogan et al., 2012b]; for example, we may select Easter Island as the canonical node and merge the edges of Rapa Nui onto it, enabling us to find, e.g., World Heritage Sites in the Pacific Ocean from Figure 8.1 based on the (consolidated) sub-graph World Heritage SitenamedEaster IslandoceanPacific.

Correction

As opposed to completion – which finds new edges in a knowledge graph – correction identifies and removes existing incorrect edges in the knowledge graph. We here divide the principal approaches for knowledge graph correction into two main lines: fact validation, which assigns a plausibility score to a given edge, typically in reference to external sources; and inconsistency repairs, which aim to resolve inconsistencies found in the knowledge graph through ontological axioms.

Fact validation

The task of fact validation (aka fact checking) [Gerber et al., 2015, Syed et al., 2018, Yin et al., 2008, Syed et al., 2019, Esteves et al., 2018, Shiralkar et al., 2017, Shi and Weninger, 2016, Socher et al., 2013, Bordes et al., 2013] involves assigning plausibility or veracity scores to facts/edges, typically between \(0\) and \(1\). An ideal fact-checking function assumes a hypothetical reference universe (an ideal knowledge graph) and would return \(1\) for the fact Santa LucíacitySantiago (being true) while returning \(0\) for SotomayorcitySantiago (being false). There is a clear relation between fact validation and link prediction – with both relying on assessing the plausibility of edges/facts/links – and indeed the same numeric- and symbol-based techniques can be applied for both cases. However, fact validation often considers online assessment of edges given as input, whereas link prediction is often an offline task that generates novel candidate edges to be assessed from the knowledge graph. Furthermore, works on fact validation are characterised by their consideration of external reference sources, which may be unstructured sources [Gerber et al., 2015, Syed et al., 2018, Samadi et al., 2016, Yin et al., 2008] or structured sources  [Syed et al., 2019, Shiralkar et al., 2017, Shi and Weninger, 2016, Socher et al., 2013, Bordes et al., 2013].

Approaches based on unstructured sources assume that they are given a verbalisation function – using, for example, rule-based approaches [Ngonga Ngomo et al., 2013, Ell et al., 2014], encoder–decoder architectures [Gardent et al., 2017], etc. – that is able to translate edges into natural language. Thereafter, approaches for computing the plausibility of facts in natural language – called fact finders [Pasternack and Roth, 2010, Pasternack and Roth, 2011] – can be directly employed. Many fact finding algorithms construct an \(n\)-partite (often bipartite) graph whose nodes are facts and sources, where a source is connected to a fact if the source “evidences” the fact, i.e., if it contains a text snippet that matches – with sufficient confidence – the verbalisation of the input edge. Two mutually-dependent scores, namely the trustworthiness of sources and the plausibility of facts, are then calculated based on this graph, where fact finders differ on how they compute these scores [Pasternack and Roth, 2011]. Here we mention three scores proposed by Pasternack and Roth [2010]

Pasternack and Roth [2011] then show that these three algorithms can be generalised into a single multi-layered graph-based framework within which (1) a source can support a fact with a weight expressing uncertainty, (2) similar facts can support each other, and (3) sources can be grouped together leading to an implicit support between sources of the same group. Other approaches for fact checking of knowledge graphs later extended this framework [Galland et al., 2010, Samadi et al., 2016]. Alternative approaches based on machine learning classifiers have also emerged, where commonly-used features include trust scores for information sources, co-occurrences of facts in sources, and so forth [Gerber et al., 2015, Syed et al., 2018].

Approaches for fact validation based on structured data typically assume external knowledge graphs as reference sources and are based on finding paths that support the edge being validated. Unsupervised approaches search for undirected [Shiralkar et al., 2017, Ciampaglia et al., 2015] or directed [Syed et al., 2019] paths up to a given threshold length that support the input edge. The relatedness between input edges and paths is computed using a mutual information function, such as normalised pointwise mutual information [Bouma, 2009]. Supervised approaches rather extract features for input edges from external knowledge graphs [Sun et al., 2011, Zhao et al., 2015, Lao and Cohen, 2010] and train a classification model to label the edges as true or false. An important set of features are metapaths, which encode sequences of predicates that correlate positively with the edge label of the input edge. Amongst such works, PredPath [Shi and Weninger, 2016] automatically extracts metapaths based on type information. Several approaches rather encode the reference nodes and edges using graph embeddings (see Section 5.2), which are then used to estimate the plausibility of the input edge being validated.

Inconsistency repairs

Ontologies can contain axioms – such as disjointness – that lead to inconsistencies. While such axioms can be provided by experts, they can can also be derived through symbolic learning, as discussed in Section 5.4. Such axioms can then be used to detect inconsistencies. With respect to correcting a knowledge graph, however, detecting inconsistencies is not enough: techniques are also required to repair such inconsistencies, which itself is not a trivial task. In the simplest case, we may have an instance of two disjoint classes, such as that Santiago is of type City and Airport, which are stated or found to be disjoint. To repair the inconsistency, it would be preferable to remove only the “incorrect” class, but which should we remove? This is not a trivial question, particularly if we consider that one edge can be involved in many inconsistencies, and one inconsistency can involve many edges. The issue of computing repairs becomes more complex when entailment is considered, where we not only need to remove the stated type, but also all of the ways in which it might be entailed; for example, removing the edge SantiagotypeAirport is insufficient if we further have an edge AricaflightSantiago combined with an axiom flightrangeAirport. Töpper et al. [2012] suggest potential repairs for such violations – remove a domain/range constraint, remove a disjointness constraint, remove a type edge, or remove an edge with a domain/range constraint – where one is chosen manually. In contrast, Bonatti et al. [2011] propose an automated method to repair inconsistencies based on minimal hitting sets [Reiter, 1987], where each set is a minimal explanation for an inconsistency. The edges to remove are chosen based on scores of the trustworthiness of their sources and how many minimal hitting sets they are either elements of or help to entail an element of, where the knowledge graph is revised to avoid re-entailment of the removed edges. Rather than repairing the data, another option is to evaluate queries under inconsistency-aware semantics, such as returning consistent answers valid under every possible repair [Lukasiewicz et al., 2013].

Other Refinement Tasks

In comparison to the quality clusters discussed in Chapter 7, the refinement methods discussed herein address particular aspects of the accuracy, coverage, and coherency dimensions. Beyond these, one could conceive of further refinement methods to address further quality issues of knowledge graphs, such as succinctness. In general, however, the refinement tasks of knowledge graph completion and knowledge graph correction have received the majority of attention until now. For further details on knowledge graph refinement, we refer to the survey by Paulheim [2017].

Publication

While it may not always be desirable to publish knowledge graphs (for example, those that offer a competitive advantage to a company [Noy et al., 2019], it may be desirable or even required to publish other knowledge graphs, such as those produced by volunteers [Vrandečić and Krötzsch, 2014, Mahdisoltani et al., 2015, Lehmann et al., 2015], by publicly-funded research [Callahan et al., 2013, Groth et al., 2014, The UniProt Consortium, 2014], by governmental organisations [Hendler et al., 2012, Shadbolt and O'Hara, 2013]. Publishing refers to making the knowledge graph (or part thereof) accessible to the public, often on the Web. Knowledge graphs published as open data are called open knowledge graphs (discussed in Section 10.1).

In the following, we first discuss two sets of principles that have been proposed to guide the publication of data on the Web. We next discuss access protocols by which the public can interact with the content of a knowledge graph. Finally, we consider techniques to restrict the access or usage of (parts of) a knowledge graph.

Best Practices

We now discuss two key sets of publishing principles: the FAIR Principles Wilkinson et al. [2016], and the Linked Data Principles [Berners-Lee, 2006].

FAIR Principles

The FAIR Principles were originally proposed in the context of publishing scientific data [Wilkinson et al., 2016] – particularly motivated by maximising the impact of publicly-funded research – but the principles generally apply to other situations where data are to be published in a manner that facilitates their re-use by external agents, with particular emphasis on machine-readability.

FAIR itself is an acronym for four foundational principles, each with particular goals [Wilkinson et al., 2016], that may apply to data, metadata, or both – the latter being denoted (meta)data.32note 32 Metadata are data about data. The distinction is often important in observational sciences, where in astronomy, for example, data may include raw image data, while metadata may include coordinates and time. We now describe the FAIR principles (slightly rephrasing the original wording in some cases for brevity [Wilkinson et al., 2016]).

In the context of knowledge graphs, a variety of vocabularies, tools, and services have been proposed that both directly and indirectly help to satisfy the FAIR principles. In terms of Findability, as discussed in Chapter 2, IRIs are built into the RDF model, providing a general schema for global identifiers. In addition, resources such as the Vocabulary of Interlinked Datasets (VoID) [Alexander et al., 2009] allow for representing metadata about graphs, while services such as DataHub [Bhardwaj et al., 2015] provide a central repository of such dataset descriptions. Access protocols that enable Accessibility will be discussed in Section 9.2, while mechanisms for authorisation will be discussed in Section 9.3. With respect to Interoperability, as discussed in Chapter 4, ontologies serve as a general knowledge representation formalism, and can in turn be used to describe vocabularies that follow FAIR principles. Regarding Reusability, licensing will be discussed in Section 9.3, while the PROV Data Model [Gil et al., 2013] discussed in Chapter 3, can encode provenance in detail.

Various knowledge graphs have been published using FAIR principles, where Wilkinson et al. [2016] explicitly mention Open PHACTS [Groth et al., 2014], a data integration platform for drug discovery, and UniProt [The UniProt Consortium, 2014], a large collection of protein sequence and annotation data, as conforming to FAIR principles. Both datasets offer graph views of their content through RDF.

Linked Data Principles

Wilkinson et al. [2016] state that FAIR Principles “precede implementation choices”, meaning that the principles do not cover how they can or should be achieved. Preceding the FAIR Principles by almost a decade are the Linked Data Principles, proposed by Berners-Lee [2006], which provide a technical basis for one possible way in which these FAIR Principles can be achieved. Specifically the Linked Data Principles are as follows:

  1. Use IRIs as names for things.
  2. Use HTTP IRIs so those names can be looked up.
  3. When a HTTP IRI is looked up, provide useful content about the entity that the IRI names using standard data formats.
  4. Include links to the IRIs of related entities in the content returned.

These principles were proposed in a Semantic Web setting, where for principle (3), the standards based on RDF (including RDFS, OWL, etc.) are currently recommended for use, particularly because they allow for naming entities using HTTP IRIs, which further paves the way for satisfying all four principles. As such, these principles outline a way in which (RDF) graph-structured data can be published on the Web such that these graphs are interlinked to form what Berners-Lee [2006] calls a “Web of Data”, whose goal is to increase automation on the Web by making content available not only in (HTML) documents intended for human consumption, but also as (RDF) structured data that machines can locate, retrieve, combine, validate, reason over, query over, etc., towards solving tasks automatically [Hogan, 2020b]. Conceptually, the Web of Data is then composed of graphs of data published on individual web-pages, where one can click on a node or edge-label – or more precisely perform a HTTP lookup on an IRI of the graph – to be transported to another graph elsewhere on the Web with relevant content for that node or edge-label, and so on recursively.

Figure 9.1 provides a small example with two Linked Data documents published on the Web, with each containing an RDF graph. As discussed in Section 3.2, terms such as clv:Concert, wd:Q142701, rdfs:label, etc., are abbreviations for IRIs, where, for example, wd:Q142701 expands to http://www.wikidata.org/entity/Q142701. Prefixes beginning with cl are fictitious prefixes we assume to have been created by the Chilean tourist board. The IRIs prefixed with \(\hookrightarrow\)Earth indicate the document returned if the node is looked up. The leftmost document is published by the tourist board and describes Lollapalooza 2018 (identified by the node cle:LP2018), which links to the headlining act Pearl Jam (wd:Q142701) described by an external knowledge graph, namely Wikidata. By looking up the node wd:Q142701 in the leftmost graph, the IRI dereferences (i.e., returns via HTTP) the document with the RDF graph on the right describing that entity in more detail. From the rightmost document, the node wd:Q221535 can be looked up, in turn, to find a graph about Eddie Vedder (not shown in the example). The IRIs for entities and documents are distinguished to ensure that we do not confuse data about the entity and the document; for example, while wd:Q221535 refers to Eddie Vedder, the IRI wdd:Q221535 refers to the document about Eddie Vedder; if we were to assign a last-modified date to the document, we should use wdd:Q221535 not wd:Q221535. In Figure 9.1, we can further observe that edge labels (which are also IRIs) and nodes representing classes (e.g., clv:Concert) can also be dereferenced, typically returning semantic definitions of the respective terms.

Two example Linked Data documents from two websites, each containing an RDF graph, where wd:Q142701 refers to Pearl Jam in Wikidata while wdd:Q142701 refers to the RDF graph about Pearl Jam, and where wd:Q221535 refers to Eddie Vedder while wdd:Q221535 refers to the RDF graph about Eddie Vedder; the edge-label wdt:571 refers to “inception” in Wikidata, while wdt:527 refers to “has part”
Two example Linked Data documents from two websites, each containing an RDF graph, where wd:Q142701 refers to Pearl Jam in Wikidata while wdd:Q142701 refers to the RDF graph about Pearl Jam, and where wd:Q221535 refers to Eddie Vedder while wdd:Q221535 refers to the RDF graph about Eddie Vedder; the edge-label wdt:571 refers to “inception” in Wikidata, while wdt:527 refers to “has part”

A key challenge is posed by the fourth principle – include links to related entities – as illustrated in Figure 9.1, where wd:Q221535 in the leftmost graph constitutes a link to related content about Pearl Jam in an external knowledge graph. Specifically, the link discovery task considers adding such links from one knowledge graph to another, which may involve inclusion of IRIs that dereference to external graphs (per Figure 9.1), or links with special semantics such as identity links. In comparison with the link prediction task discussed in Section 8.1, which is used to complete links within a knowledge graph, link discovery aims to discover links across knowledge graphs, which involves unique aspects: first, link discovery typically considers disjoint sets of source (local) nodes and target (remote) nodes; second, the knowledge graphs may often use different vocabularies; third, while in link prediction there already exist local examples of the links to predict, in link discovery, there are often no existing links between knowledge graphs to learn from. A common technique is to define manually-crafted linkage rules (aka link specifications) that apply heuristics for defining links that potentially incorporate similarity measures [Ngonga Ngomo and Auer, 2011, Volz et al., 2009]. Link discovery is greatly expedited by the provision of standard identifier schemes within knowledge graphs, such as ISBNs for books, alpha-2 and alpha-3 codes for countries (e.g., cl, clp), or even links to common knowledge graphs such as DBpedia [Lehmann et al., 2015] or Wikidata [Vrandečić and Krötzsch, 2014] (that themselves include standard identifiers). We refer to the survey on link discovery by Nentwig et al. [2017] for more details.

More finer-grained recommendations for publishing Linked Data have also been proposed, relating to how best to implement dereferencing, what kinds of links to include, how to publish and interlink vocabularies, amongst other considerations [Heath and Bizer, 2011, Janowicz et al., 2014]. We refer to the book by Heath and Bizer [2011] for more discussion on how to publish Linked Data on the Web.

Access Protocols

Publishing involves giving access to the public to interact with the knowledge graph, which implies the provision of access protocols that define the requests that agents can make and the response that they can expect as a result. Per the Accessibility principle of FAIR (specifically A1.1), this protocol should be open, free, and universally implementable. In the context of knowledge graphs, as shown in Figure 9.2, there are a number of access protocols to choose from, varying from simple protocols that allow users to simply download all content, towards protocols that accept and evaluate increasingly complex requests. While simpler protocols require less computation on the server that publishes the data, more complex protocols allow agents to request more specific data, thus reducing bandwidth. A knowledge graph may also offer a variety of access protocols catering to different agents with different requirements [Verborgh et al., 2014]. We now discuss such access protocols.

Access protocols for knowledge graphs, from simple protocols (left) to more complex protocols (right)
Access protocols for knowledge graphs, from simple protocols (left) to more complex protocols (right)

Dumps

A dump is a file or collection of files containing the content of the knowledge graph available for download. The request in this case is for the file(s) and the response is the content of the file(s). In order to publish dumps, first of all, concrete – and ideally standard – syntaxes are required to serialise the graph. While for RDF graphs there are various standard syntaxes available based on XML [Gandon and Schreiber, 2014], JSON [Sporny et al., 2014], custom syntaxes [Prud'hommeaux and Carothers, 2014], and more besides, currently there are only non-standard syntaxes available for property graphs [Tomaszuk et al., 2019]. Second, to reduce bandwidth, compression methods can be applied. While standard compression such as GZIP or BZip2 can be straightforwardly applied on any file, custom compression methods have been proposed for graphs that not only offer better compression ratios than these standard methods, but also offer additional functionalities, such as compact indexes for performing efficient lookups once the file is downloaded [Fernández et al., 2013]. Finally, to further reduce bandwidth, when the knowledge graph is updated, “diffs” can be computed and published to obviate the need for agents to download all data from scratch (see [Tummarello et al., 2007, Papavasileiou et al., 2013, Ahn et al., 2015]). Still, however, dumps are only suited to certain use-cases, in particular for agents that wish to maintain a full local copy of a knowledge graph. If an agent were rather only interested in, for example, all food festivals in Santiago, downloading the entire dump may require transferring and processing a lot of irrelevant data.

Node lookups

Protocols for performing node lookups accept a node (id) request (e.g., cle:LP2018 in Figure 9.1) and return a (sub-)graph describing that node (e.g., the document cld:LP2018). Such a protocol is the basis for the Linked Data principles outlined previously, whereby node lookups are implemented through HTTP dereferencing, which further allows nodes in remote graphs to be referenced from across the Web. Although there are varying definitions on what content should be returned for a node [Stickler, 2005], a common convention is to return a sub-graph containing either all outgoing edges for that node or all incident edges (both outgoing and incoming) for that node [Hogan et al., 2012a]. Though simple, mechanisms for evaluating graph patterns can be implemented on top of a node lookup interface by traversing from node to node per the particular graph pattern [Hartig et al., 2009]; for example, to find all food festivals in Santiago – represented by the graph pattern Food Festivaltype?fflocationSantiago – we may perform a node lookup for Santiago, subsequently performing a node lookup for each node connected by a location edge to Santiago, returning those nodes declared to be of type Food Festival. However, such an approach may not be feasible if no starting node is declared (e.g., if all nodes are variables), if the node lookup service does not return incoming edges, etc. The client agent may also need to request more data than necessary; for example, the document returned for Santiago may return a lot of data irrelevant to the query, and nodes with a location in Santiago that are not instances of Food Festival still need to be looked up to check their type. Node lookups are relatively inexpensive for servers to support in terms of CPU, but may again waste bandwidth due to transferring irrelevant data.

Edge patterns

Edge patterns – also known as triple patterns in the case of directed, edge-labelled graphs – are singleton graph patterns, i.e., graph patterns with a single edge. Examples of edge patterns are ?fftypeFood Festival or ?fflocationSantiago, etc., where any term can be a variable or a constant. A protocol for edge patterns accepts such a pattern and returns all solutions for the pattern. Edge patterns provide more flexibility than node lookups, where graph patterns are more readily decomposed into edge patterns than node lookups. With respect to the agent interested in food festivals in Santiago, they can first, for example, request solutions for the edge pattern ?fflocationSantiago and locally join/intersect these solutions with those of ?fftypeFood Festival. Given that some edge patterns (e.g., ?x?y?z) can return many solutions, protocols for edge patterns may offer additional practical features such as iteration or pagination over results [Verborgh et al., 2016]. Much like node lookups, the server cost of responding to a request is relatively low and easy to predict. However, the server may often need to transfer irrelevant intermediate results to the client, which in the previous example may involve returning nodes located in Santiago that are not food festivals. This issue is further aggravated if the client does not have access to statistics about the knowledge graph in order to plan how to best perform the join; for example, if there are relatively few food festivals but many things located in Santiago, rather than intersecting the solutions of the two aforementioned edge patterns, it should be more efficient to send a request for each food festival to see if it is in Santiago, but deciding this requires statistics about the knowledge graph. Extensions to the edge-pattern protocol have thus been proposed to allow for more efficient joins [Hartig et al., 2017], such as allowing batches of solutions to be sent alongside the edge pattern to only return solutions compatible with the solutions in the request [Hartig and Buil Aranda, 2016] (e.g., sending a batch of solutions for ?fftypeFood Festival to join with the solutions for the request ?fflocationSantiago).

(Complex) graph patterns

Another alternative is to let client agents make requests based on (complex) graph patterns (see Section 2.2), with the server returning (only) the final solutions. In our running example, this involves the client issuing a request for Food Festivaltype?fflocationSantiago and directly receiving the relevant results. Compared with the previous protocols, this protocol is much more efficient in terms of bandwidth: it allows clients to make more specific requests and the server to return more specific responses. However, this reduction in bandwidth use comes at the cost of the server having to evaluate much more complex requests, where, furthermore, the costs of a single request are much more difficult to anticipate. While a variety of optimised engines exist for evaluating (complex) graph patterns (e.g., [Erling, 2012, Miller, 2013, Thompson et al., 2014] amongst many others), the problem of evaluating such queries is known to be intractable [Angles et al., 2017]. Perhaps for this reason, public services offering such a protocol (most often supporting SPARQL queries [Harris et al., 2013]) have been found to often exhibit downtimes, timeouts, partial results, slow performance, etc. [Buil-Aranda et al., 2013b]. Even considering such issues, however, popular services continue to receive – and successfully evaluate – millions of requests/queries per day [Malyshev et al., 2018, Saleem et al., 2015], with difficult (worst-case) instances being rare in practice [Bonifati et al., 2017].

Other protocols

While Figure 9.2 makes explicit reference to some of the most commonly-encountered access protocols found for knowledge graphs in practice, one may of course imagine other protocols lying almost anywhere on the spectrum from more simple to more complex interfaces. To the right of (Complex) Graph Patterns, one could consider supporting even more complex requests, such as queries with entailments [Glimm, 2011], queries that allow recursion [Reutter et al., 2015], federated queries that can join results from remote services [Buil-Aranda et al., 2013a], or even (hypothetically) supporting Turing-complete requests that allow running arbitrary procedural code on a knowledge graph. As mentioned at the outset, a server may also choose to support multiple, complementary protocols [Verborgh et al., 2014].

Usage Control

Considering our hypothetical tourism knowledge graph, at first glance, one might assume that the knowledge required to deliver the envisaged services is public and thus can be used both by the tourism board and the tourists. On closer inspection, however, we may see the need for usage control in various forms:

Thus in this section, we examine the state of the art in terms of knowledge graph licensing, usage policies, encryption, and anonymisation.

Licensing

When it comes to associating machine readable licenses with knowledge graphs, the W3C Open Digital Rights Language (ODRL) [Iannella and Villata, 2018] provides an information model and related vocabularies that can be used to specify permissions, duties, and prohibitions with respect to actions relating to assets. ODRL supports fine-grained descriptions of digital rights that are represented as – and thus can be embedded within – graphs. Figure 9.3 illustrates a license granting the assignee the permission to Modify, Distribute, and Derive work from the Event Graph (e.g., Figure 2.1); however the assignee is obliged to Attribute the copyright holder. From a modelling perspective, ODRL can be used to model several well-known license families, for instance Apache, Creative Commons (CC), and Berkeley Software Distribution (BSD), to name but a few [Cabrio et al., 2014, Panasiuk et al., 2018]. Additionally, Cabrio et al. [2014] propose methods to automatically extract machine-readable licenses from unstructured text. From a reasoning perspective, license compatibility validation and composition techniques [Villata and Gandon, 2012, Governatori et al., 2013, Moreau et al., 2019] can be used to combine knowledge graphs that are governed by different licenses. Such techniques are employed by the the Data Licenses Clearance Center (DALICC), which includes a library of standard machine readable licenses, and tools that enable users both to compose arbitrary custom licenses and also to verify the compatibility of different licenses [Pellegrini et al., 2019].

A license for event data, along with permissions, actions, and obligations
A license for event data, along with permissions, actions, and obligations

Usage policies

Access control policies based on edge patterns can be used to restrict access to parts of a knowledge graph [Reddivari et al., 2005, Flouris et al., 2010, Kirrane et al., 2013]. WebAccessControl (WAC)33note 33 WAC, http://www.w3.org/wiki/WebAccessControl is an access control framework for graphs that uses WebID for authentication and provides a vocabulary for specifying access control policies. Extensions of this WAC vocabulary have been proposed to capture privacy preferences [Sacco and Passant, 2011] and to cater for contextual constraints [Villata et al., 2011, Costabello et al., 2012]. Although ODRL is primarily used to specify licenses, profiles to additionally specify access policies [Steyskal and Polleres, 2014] and regulatory obligations [Agarwal et al., 2018, De Vos et al., 2019] have also been proposed in recent years, as discussed in the survey by Kirrane et al. [2017].

As a generalisation of access policies, usage policies specify how data can be used: what kinds of processing can be applied, by whom, for what purpose, etc. The example usage policy presented in Figure 9.4 states that the process Analyse of Location Graph can be performed on Internal Servers by members of Company Staff in order to provide Event Recommendations. Vocabularies for usage policies have been proposed by the SPECIAL H2020 project [Bonatti et al., 2019] and the W3C Data Privacy Vocabularies and Controls Community Group (DPVCG) [Pandit et al., 2019, Bonatti and Kirrane, 2019]. Once specified in these vocabularies, usage policies can then be used to verify that data processing conforms to legal norms and to the consent provided by subjects [Delanaux et al., 2018, Bonatti and Kirrane, 2019].

A policy for usage of a sub-graph of location data in the knowledge graph
A policy for usage of a sub-graph of location data in the knowledge graph

Encryption

Rather than internally controlling usage, the tourist board could use encryption mechanisms on parts of the published knowledge graph, for example relating to reports of crimes, and provide keys to partners who should have access to the plaintext. While a straightforward approach is to encrypt the entire graph (or sub-graphs) with one key, more fine-grained encryption can be performed for individual nodes or edge-labels in a graph, potentially providing different clients access to different information through different keys [Giereth, 2005]. The CryptOntology [Gerbracht, 2008] can further be used to embed details about the encryption mechanism used within the knowledge graph. Figure 9.5 illustrates how this could be used to encrypt the names of claimants from Figure 6.4, storing the ciphertext zhk…kjg, as well as the key-length and encryption algorithm used. In order to grant access to the plaintext, one approach is to encrypt individual edges with symmetric keys so as to allow specific types of edge patterns to only be executed by clients with the appropriate key [Kasten et al., 2013]. This approach can be used, for example, to allow clients who know a claimant ID (e.g., Claimant-XY12SDA) and have the appropriate key to find (only) the name of the claimant through an edge pattern Claimant-XY12SDAClaimant-name?name. A key limitation of this approach, however, is that it requires attempting to decrypt all edges to find all possible solutions. A more efficient alternative is to combine functional encryption and specialised indexing to retrieve solutions from the encrypted graph without attempting to decrypt all edges [Fernández et al., 2017].

Directed edge-labelled graph with the name of the claimant encrypted; plaintext elements are dashed and may be omitted from published data (possibly along with encryption details)
Directed edge-labelled graph with the name of the claimant encrypted; plaintext elements are dashed and may be omitted from published data (possibly along with encryption details)

Anonymisation

Consider that the tourist board acquires information on transport taken by individuals within the country, which can be used – not only by the board, but potentially other stakeholders, such as travel companies – to understand trajectories taken by tourists. However, from a data-protection perspective, it would be advisable to anonymise the knowledge graph to avoid leaking the personal travel history of individuals.

A first approach to anonymisation is to suppress and generalise knowledge in a graph such that individuals cannot be identified, based on \(k\)-anonymity [Samarati and Sweeney, 1998]34note 34 \(k\)-anonymity guarantees that the data of an individual is indistinguishable from at least \(k-1\) other individuals., \(l\)-diversity [Li et al., 2007]35note 35 \(l\)-diversity guarantees that sensitive data fields have at least \(l\) diverse values within each group of individuals; this avoids leaks such as that all tourists from Austria (a group of individuals) in the data have been pick-pocketed (a sensitive attribute), which would reveal sensitive information about individuals from Austria., etc. Approaches that apply \(k\)-anonymity on graphs identify and suppress “quasi-identifiers” that would allow a given individual to be distinguished from fewer than \(k-1\) other individuals [Radulovic et al., 2015, Heitmann et al., 2017]. Figure 9.6 illustrates a possible result of \(k\)-anonymisation for a sub-graph describing a flight passenger, where quasi-identifiers (passport, plane ticket) have been converted into blank nodes, ensuring that the passenger (the dashed blank node) cannot be distinguished from \(k-1\) other individuals. In the context of a graph, however, neighbourhood attacks [Zhou and Pei, 2011] – using information about neighbours – can also break \(k\)-anonymity, where we also suppress the day and time of the flight, which, though not sensitive information per se, could otherwise break \(k\)-anonymity for passengers (if, for example, a particular flight had fewer than \(k\) males from the U.S. onboard). The graph shown in Figure 9.6 then offers \(k\)-anonymity for the particular individual assuming that at least \(k\) male passengers from the U.S. flew during December 2018 from Arica to Santiago.

Anonymised sample of a directed edge-labelled graph describing a passenger (dashed) of a flight
Anonymised sample of a directed edge-labelled graph describing a passenger (dashed) of a flight

More complex neighbourhood attacks may rely on more abstract graph patterns, observing that individuals can be deanonymised purely from knowledge of the graph structure, even if all nodes and edge labels are left blank; for example, if we know that a team of \(k-1\) players take flights together for a particular number of away games, we could use this information for a neighbourhood attack that reveals the set of players in the graph. Hence a number of guarantees specific to graphs have been proposed, including \(k\)-degree anonymity [Liu and Terzi, 2008], which ensures that individuals cannot be deanonymised by attackers with knowledge of the degree of particular individuals. The approach is based on minimally modifying the graph to ensure that each node has at least \(k-1\) other nodes with the same degree. A stronger guarantee, called \(k\)-isomorphic neighbour anonymity [Zhou and Pei, 2008], avoids neighbourhood attacks where an attacker knows how an individual is connected to nodes in their neighbourhood; this is done by modifying the graph to ensure that for each node, there exist at least \(k-1\) nodes with isomorphic (i.e., identically structured) neighbourhoods elsewhere in the graph. Both approaches only protect against attackers with knowledge of bounded neighbourhoods. An even stronger notion is that of \(k\)-automorphism [Zou et al., 2009], which ensures that for every node, it is structurally indistinguishable from \(k-1\) other nodes, thus avoiding any attack based on structural information (as a trivial example, a \(k\)-clique or a \(k\)-cycle satisfy \(k\)-automorphism). Many of these techniques for anonymisation of graph data were motivated by social networks [Narayanan and Shmatikov, 2009], though they can also be applied to knowledge graphs, per the work of Lin and Tripunitara [2017], who adapt \(k\)-automorphism for directed edge-labelled graphs (specifically RDF graphs).

While the aforementioned approaches anonymise data, a second approach is to apply anonymisation when answering queries, such as adding noise to the solutions in a way that preserves privacy. One approach is to apply \(\varepsilon\)-differential privacy [Dwork, 2006]36note 36 \(\varepsilon\)-differential privacy ensures that the probability of a given result from a process (e.g., query) applied to data, to which random noise is added, differs no more than \(e^\varepsilon\) when the data includes or excludes any individual. for querying graphs [Silva et al., 2017]. Such mechanisms are typically used for aggregate (e.g., count) queries, where noise is added to avoid leaks about individuals. To illustrate, differential privacy may allow for counting the number of passengers of specified nationalities taking specified flights, adding (just enough) random noise to the count to ensure that we cannot tell, within a certain probability (controlled by \(\varepsilon\)), whether or not a particular individual took a flight, where, intuitively speaking, we would require (proportionally) less noise for nationalities with many passengers in the data, but more noise to “hide” passengers from more uncommon nationalities.

These approaches require information loss for stronger guarantees of privacy; which to choose is thus heavily application dependent. If the anonymised data are to be published in their entirety as a “dump”, then an approach based on \(k\)-anonymity can be used to protect individuals, while \(l\)-diversity can be used to protect groups. On the other hand, if the data are to be made available, in part, through a query interface, then \(\varepsilon\)-differential privacy is a more suitable framework.

Knowledge Graphs in Practice

In this chapter, we discuss some of the most prominent knowledge graphs that have emerged in the past years. We begin by discussing open knowledge graphs, most of which have been published on the Web per the guidelines and protocols described in Chapter 9. We later discuss enterprise knowledge graphs that have been created by companies from diverse industries for a wide range of applications.

Open Knowledge Graphs

By open knowledge graphs, we refer to knowledge graphs published under the Open Data philosophy, namely that “open means anyone can freely access, use, modify, and share for any purpose (subject, at most, to requirements that preserve provenance and openness)”.37note 37 See http://opendefinition.org/ Many open knowledge graphs have been published in the form of Linked Open Datasets [Heath and Bizer, 2011], which are (RDF) graphs published under the Linked Data principles (see Section 9.1.2) following the Open Data philosophy. Many of the most prominent open knowledge graphs – including DBpedia [Lehmann et al., 2015], YAGO [Suchanek et al., 2007], Freebase [Bollacker et al., 2007b], and Wikidata [Vrandečić and Krötzsch, 2014] – cover multiple domains, representing a broad diversity of entities and relationships; we first discuss these in turn. Later we discuss some of the other (specific) domains for which open knowledge graphs are currently available. Most of the open knowledge graphs we discuss in this section are modelled in RDF, published following Linked Data principles, and offer access to their data through dumps (RDF), node lookups (Linked Data), graph patterns (SPARQL) and, in some cases, edge patterns (Triple Pattern Fragments).

DBpedia

The DBpedia project was developed to extract a graph-structured representation of the semi-structured data embedded in Wikipedia articles [Auer et al., 2007], enabling the integration, processing, and querying of these data in a unified manner. The resulting knowledge graph is further enriched by linking to external open resources, including images, webpages, and external datasets such as DailyMed, DrugBank, GeoNames, MusicBrainz, New York Times, and WordNet [Lehmann et al., 2015]. The DBpedia extraction framework consists of several components, corresponding to abstractions of Wikipedia article sources, graph storage and serialisation destinations, wiki-markup extractors, parsers, and extraction managers [Bizer et al., 2009]. Specific extractors are designed to process labels, abstracts, interlanguage links, images, redirects, disambiguation pages, external links, internal pagelinks, homepages, categories, and geocoordinates. The content in the DBpedia knowledge graph is not only multidomain, but also multilingual: as of 2012, DBpedia contained labels and abstracts in up to 97 different languages [Mendes et al., 2012a]. Entities within DBpedia are classified using four different schemata in order to address varying requirements [Bizer et al., 2009]. These schemata include a Simple Knowledge Organization System (SKOS) representation of Wikipedia categories, a Yet Another Great Ontology (YAGO) classification schema (discussed presently), an Upper Mapping and Binding Exchange Layer (UMBEL) ontology categorisation schema, and a custom schema called the DBpedia ontology with classes such as Person, Place, Organisation, and Work [Lehmann et al., 2015]. DBpedia also supports live synchronisation in order to remain consistent with dynamic Wikipedia articles [Lehmann et al., 2015].

Yet Another Great Ontology

YAGO likewise extracts graph-structured data from Wikipedia, which are then unified with the hierarchical structure of WordNet to create a “light-weight and extensible ontology with high quality and coverage” [Suchanek et al., 2007]. This knowledge graph aims to be applied for various information technology tasks, such as machine translation, word sense disambiguation, query expansion, document classification, data cleaning, information integration, etc. While earlier approaches automatically extracted structured knowledge from text using pattern matching, natural language processing (NLP), and statistical learning, the resulting content tended to lack in quality when compared with what was possible through manual construction [Suchanek et al., 2007]. However, manual construction is costly, making it challenging to achieve broad coverage and keep the data up-to-date. In order to extract data with high coverage and quality, YAGO (like DBpedia) mostly extracts data from Wikipedia infoboxes and category pages, which contain core entity information and lists of articles for a specific category, respectively. These, in turn, are unified with hierarchical concepts from WordNet [Suchanek et al., 2008]. A schema – called the YAGO model – provides a vocabulary defined in RDFS; this model allows for representing words as entities, capturing synonymy and ambiguity [Suchanek et al., 2007]. The model further supports reification, \(n\)-ary relations, and data types [Suchanek et al., 2008]. Refinement mechanisms employed within YAGO include canonicalisation, where each edge and node is mapped to a unique identifier and duplicate elements are removed, and type checking, where nodes that cannot be assigned to a class by deductive or inductive methods are eliminated [Suchanek et al., 2008]. YAGO would be extended in later years to support spatio-temporal context [Hoffart et al., 2011] and multilingual Wikipedias [Mahdisoltani et al., 2015].

Freebase

Freebase was a general-purpose, broad collection of human knowledge that aimed to address some of the large-scale information integration problems associated with the decentralised nature of the Semantic Web, such as uneven adoption, implementation challenges, and distributed query performance limitations [Bollacker et al., 2007a]. Unlike DBpedia and YAGO – which are mostly extracted from Wikipedia/WordNet – Freebase solicited contributions directly from human editors. Included in the Freebase platform were a scalable data store with versioning mechanisms; a large data object store (LOB) for the storage of text, image, and media files; an API that could be queried using the Metaweb Query Language (MQL); a Web user interface; and a lightweight typing system [Bollacker et al., 2007a]. The latter typing system was designed to support collaborative processes. Rather than forcing ontological correctness or logical consistency, the system was implemented as a loose collection of structuring mechanisms – based on datatypes, semantic classes, properties, schema definitions, etc. – that allowed for incompatible types and properties to coexist simultaneously [Bollacker et al., 2007a]. Content could be added to Freebase interactively through the Web user interface or in an automated way by leveraging the API’s write functionality. Freebase had been acquired by Google in 2010, where the content of Freebase formed an important part of the Google Knowledge Graph announced in 2012 [Singhal, 2012]. When Freebase became read-only as of March 2015, the knowledge graph contained over three billion edges. Much of this content was subsequently migrated to Wikidata [Pellissier Tanon et al., 2016].

Wikidata

Wikipedia contains a wealth of semi-structured data embedded in info-boxes, lists, tables, etc., as exploited by DBpedia and YAGO. However, these data have traditionally been curated and updated manually across different articles and languages; for example, a goal scored by a Chilean football player may require manual updates in the player's article, the tournament article, the team article, lists of top scorers, and so forth, across hundreds of language versions. Manual curation has led to a variety of data quality issues, including contradictory data in different articles, languages, etc. The Wikimedia Foundation uses Wikidata as a centralised, collaboratively-edited knowledge graph to supply Wikipedia – and arbitrary other clients – with data. Under this vision, a fact could be added to Wikidata once, triggering the automatic update of potentially multitudinous articles in Wikipedia across different languages [Vrandečić and Krötzsch, 2014]. Like Wikipedia, Wikidata is also considered a secondary source containing claims that should reference primary sources, though claims can also be initially added without reference [Piscopo et al., 2017]. Wikidata further allows for different viewpoints in terms of potentially contradictory (referenced) claims [Vrandečić and Krötzsch, 2014]. Wikidata is multilingual, where nodes and edges are assigned language-agnostic Qxx and Pxx codes (see Figure 9.1) and are subsequently associated with labels, aliases, and descriptions in various languages [Kaffee et al., 2017], allowing claims to be surfaced in these languages. Collaborative editing is not only permitted on the data level, but also on the schema level, allowing users to add or modify lightweight semantic axioms [Piscopo and Simperl, 2018] – including sub-classes, sub-properties, inverse properties, etc. – as well as shapes [Boneva et al., 2019]. Wikidata offers various access protocols [Malyshev et al., 2018] and has received broad adoption, being used by Wikipedia to generate infoboxes in certain domains [Sáez and Hogan, 2018], being supported by Google [Pellissier Tanon et al., 2016], and having been used as a data source for prominent end-user applications such as Apple’s Siri, amongst others [Malyshev et al., 2018].

Other open cross-domain knowledge graphs

Aside from DBpedia, YAGO, Freebase and Wikidata, a number of other cross-domain knowledge graphs have been developed down through the years. BabelNet [Navigli and Ponzetto, 2012], like YAGO, is based on unifying WordNet and Wikipedia, but with the integration of additional knowledge graphs such as Wikidata, and a focus on creating a knowledge graph of multilingual lexical forms (organised into multilingual synsets) by transforming lexicographic resources such as Wiktionary and OmegaWiki into knowledge graphs. Compared to other knowledge graphs, lexicalised knowledge graphs such as BabelNet bring together the encyclopedic information found in Wikipedia with the lexicographic information usually found in monolingual and bilingual dictionaries. The Cyc project [Lenat, 1995] aims to encode common-sense knowledge in a machine-readable way, where over 900 person-years of effort [Matuszek et al., 2006] have, since 1986, gone into the creation of 2.2 million facts and rules. Though Cyc is proprietary, an open subset called OpenCyc has been published, where we refer to the comparison by Färber et al. [2018] of DBpedia, Freebase, OpenCyc, and YAGO for further details. The Never Ending Language Learning (NELL) project [Mitchell et al., 2018] has, since 2010, extracted a graph of 120 million edges from the text of web pages using OIE methods (see Chapter 6). Each such open knowledge graph applies different combinations of the languages and techniques discussed in this book over different sources with differing results.

Domain-specific open knowledge graphs

Open knowledge graphs have been published in a variety of specific domains. Schmachtenberg et al. [2014] identify the most prominent domains in the context of Linked Data as follows: media, relating to news, television, radio, etc. (e.g., the BBC World Service Archive [Raimond et al., 2014]); government, relating to the publication of data for transparency and development (e.g., by the U.S. [Hendler et al., 2012] and U.K. [Shadbolt and O'Hara, 2013] governments); publications, relating to academic literature in various disciplines (e.g., OpenCitations [Peroni et al., 2017], SciGraph [Iana et al., 2019], Microsoft Academic Knowledge Graph [Färber, 2019]); geographic, relating to places and regions of interest (e.g., LinkedGeoData [Stadler et al., 2012]); life sciences, relating to proteins, genes, drugs, diseases, etc. (e.g., Bio2RDF [Callahan et al., 2013]); and user-generated content, relating to reviews, open source projects, etc. (e.g., Revyu [Heath and Motta, 2008]). Open knowledge graphs have also been published in other domains, including cultural heritage [Hyvönen et al., 2009], music [Raimond et al., 2009], law [Montiel-Ponsoda et al., 2017], theology [Sherif and Ngonga Ngomo, 2015], and even tourism [Lu et al., 2016, Kärle et al., 2018, Maturana et al., 2018, Zhang et al., 2019]. The envisaged applications for such knowledge graphs are as varied as the domains from which they emanate, but often relate to integration [Raimond et al., 2009, Callahan et al., 2013], recommendation [Raimond et al., 2009, Lu et al., 2016], transparency [Hendler et al., 2012, Shadbolt and O'Hara, 2013], archiving [Hyvönen et al., 2009, Raimond et al., 2014], decentralisation [Heath and Motta, 2008], multilingual support [Sherif and Ngonga Ngomo, 2015], regulatory compliance [Montiel-Ponsoda et al., 2017], etc.

Enterprise Knowledge Graphs

A variety of companies have announced the creation of proprietary “enterprise knowledge graphs” with a variety of goals in mind, which include: improving search capabilities [Singhal, 2012, Shrivastava, 2017, Krishnan, 2018, Chang, 2018, Hamad et al., 2018], providing user recommendations [Chang, 2018, Hamad et al., 2018], implementing conversational/personal agents [Pittman et al., 2017], enhancing targeted advertising [He et al., 2016], empowering business analytics [He et al., 2016], connecting users [He et al., 2016, Noy et al., 2019], extending multilingual support [He et al., 2016], facilitating research and discovery [Bendtsen and Petrovski, 2019], assessing and mitigating risk [Tobin, 2017, Dalgliesh, 2016], tracking news events [Meij, 2019], and increasing transport automation [Henson et al., 2019], amongst (many) others. Though highly diverse, these enterprise knowledge graphs do follow some high-level trends, as reflected in the discussion by Noy et al. [2019]: (1) data are typically integrated into the knowledge graph from a variety of both external and internal sources (often involving text); (2) the enterprise knowledge graph is often very large, with millions or even billions of nodes and edges, posing challenges in terms of scalability; (3) refinement of the initial knowledge graph – adding new links, consolidating duplicate entities, etc. – is important to improve quality; (4) techniques to keep the knowledge graph up-to-date with the domain are often crucial; (5) a mix of ontological and machine learning representations are often combined or used in different situations in order to draw conclusions from the enterprise knowledge graph; (6) the ontologies used tend to be lightweight, often simple taxonomies representing a hierarchy of classes or concepts. We now discuss the main industries in which enterprise knowledge graphs have been deployed.

Web search engines have traditionally focused on matching a query string with sub-strings in web documents. The Google Knowledge Graph [Singhal, 2012, Noy et al., 2019] rather promoted a paradigm of “things not strings” – analogous to semantic search [Guha et al., 2003] – where the search engine would now try to identify the entities that a particular search may be expressing interest in. The knowledge graph itself describes these entities and how they interrelate. One of the main user-facing applications of the Google Knowledge Graph is the “Knowledge Panel”, which presents a pane on the right-hand side of (some) search results describing the principal entity that the search appears to be seeking, including some images, attribute–value pairs, and a list of related entities that users also search for. The Google Knowledge Graph was key to popularising the modern usage of the phrase “knowledge graph” (see Appendix A). Other major search engines, such as Microsoft Bing38note 38 Microsoft’s Knowledge Graph was previously called “Satori” (meaning understanding in Japanese). [Shrivastava, 2017], would later announce knowledge graphs along similar lines.

Commerce

Enterprise knowledge graphs have also been announced by companies that are principally concerned with selling or renting goods and services. A prominent example of such a knowledge graph is that used by Amazon [Krishnan, 2018, Dong, 2019], which describes the products on sale in their online marketplace. One of the main stated goals of this knowledge graph is to enable more advanced (semantic) search features for products, as well as to improve product recommendations to users of its online marketplace. Another knowledge graph for commerce was announced by eBay [Pittman et al., 2017], which encodes product descriptions and shopping behaviour patterns, and is used to power conversational agents that help users to find relevant products through a natural language interface. Airbnb [Chang, 2018] has also described a knowledge graph that encodes accommodation for rent, places, events, experiences, neighbourhoods, users, tags, etc., on top of which a taxonomic schema is defined. This knowledge graph is used to offer potential clients recommendations of attractions, events, and activities available in the neighbourhood of a particular home for rent. Uber [Hamad et al., 2018] has similarly announced a knowledge graph focused on food and restaurants for their “Uber Eats” delivery service. The goals are again to offer semantic search features and recommendations to users who are uncertain of precisely what kind of food they are looking for.

Social networks

Enterprise knowledge graphs have also emerged in the context of social networking services. Facebook [Noy et al., 2019] has gathered together a knowledge graph describing not only social data about users, but also the entities they are interested in, including celebrities, places, movies, music, etc., in order to connect people, understand their interests, and provide recommendations. LinkedIn [He et al., 2016] announced a knowledge graph containing users, jobs, skills, companies, places, schools, etc., on top of which a taxonomic schema is defined. The knowledge graph is used to provide multilingual translations of important concepts, to improve targeted advertising, to provide advanced features for job search and people search, and likewise to provide recommendations matching jobs to people (and vice versa). Another knowledge graph has been created by Pinterest [Gon\c{c}alves et al., 2019], describing users and their interests, the latter being organised into a taxonomy. The main use-cases for the knowledge graph are to help users to more easily find content of interest to them, as well as to enhance revenue through targeted advertisements.

Finance

The financial sector has also seen deployment of enterprise knowledge graphs. Amongst these, Bloomberg [Meij, 2019] has proposed a knowledge graph that powers financial data analytics, including sentiment analysis for companies based on current news reports and tweets, a question answering service, as well as detecting emerging events that may affect stock values. Thomson Reuters (Refinitiv) [Tobin, 2017] has likewise announced a knowledge graph encoding “the financial ecosystem” of people, organisations, equity instruments, industry classifications, joint ventures and alliances, supply chains, etc., using a taxonomic schema to organise these entities. Some of the applications they mention for the knowledge graph include supply chain monitoring, risk assessment, and investment research. Knowledge graphs have also been used for deductive reasoning, with Banca d’Italia [Bellomarini et al., 2019] using rule-based reasoning to determine, for example, the percentage of ownership of a company by various stakeholders. Other companies exploring financial knowledge graphs include Accenture [Okorafor and Ray, 2019], Capital One [Branum and Sehon, 2019], Wells Fargo [Newman, 2019], amongst various others.

Other industries

Enterprises have also been actively developing knowledge graphs to enable novel applications in a variety of other industries, including: healthcare, where IBM are exploring use-cases for drug discovery [Noy et al., 2019] and information extraction from package inserts [Gentile et al., 2019], while AstraZeneca [Bendtsen and Petrovski, 2019] are using a knowledge graph to advance genomics research and disease understanding; transport, where Bosch are exploring a knowledge graph of scenes and locations for driving automation [Henson et al., 2019]; oil & gas, where Maana [Dalgliesh, 2016] are using knowledge graphs to perform data integration for risk mitigation regarding oil wells and drilling; and more besides.

Summary and Conclusion

We have provided a comprehensive introduction to knowledge graphs, which have been receiving more and more attention in recent years. Under the definition of a knowledge graph as a graph of data intended to accumulate and convey knowledge of the real world, whose nodes represent entities of interest and whose edges represent relations between these entities, we have discussed models by which data can be structured as graphs; representations of schema, identity and context; techniques for leveraging deductive and inductive knowledge; methods for the creation, enrichment, quality assessment and refinement of knowledge graphs; principles and standards for publishing knowledge graphs; and finally, we have discussed the adoption of both open and enterprise knowledge graphs in the real world.

In this final chapter, we provide some concluding remarks, and further offer some insights on potential future directions for research on knowledge graphs.

Concluding remarks. Knowledge graphs have garnered significant attention not only from diverse organisations and industries, but also diverse research communities. This attention is due, in no small part, to the ubiquitous nature of the problem that knowledge graphs address: integrating and extracting value from diverse sources of data at large scale, be it in the context of a particular organisation, community, or more general collections of human knowledge. The key insight of knowledge graphs is that graphs provide a simple, flexible, intuitive and yet powerful abstraction for representing and integrating diverse data at large scale. This insight is far from new (see Appendix A), but rather has finally come of age with the advent of knowledge graphs. Graphs have long been used to represent data and knowledge in areas such as Graph Algorithms and Theory, Graph Databases, Information Extraction, Knowledge Representation, Machine Learning, the Semantic Web, and more besides. The advances in these areas can now be unified and applied for knowledge graphs.

Thus, the decision to model data as a graph opens up a “tool-box” of languages, techniques and systems – stemming from diverse areas – that can be deployed in order to integrate and extract value from data at large scale, as follows:

As we have discussed in Chapter 10, the various components of this “knowledge graph tool-box” can already be found deployed in practice, having been applied – to varying degrees – in the context of numerous open and enterprise knowledge graphs. As adoption of knowledge graphs continues, work will also continue on improving and combining these tools, as well as on developing novel tools that help to better integrate and extract value from diverse sources of data at large scale.

Future directions. Research on knowledge graphs involves a confluence of techniques from different research areas with the common objective of maximising the knowledge – and thus value – that can be distilled from diverse sources at large scale using a graph-based data abstraction [Hogan, 2020a].

In the intersection of data graphs and deductive knowledge, we emphasise emerging topics such as formal semantics for property graphs, with languages that can take into account the meaning of labels and property–value pairs on nodes and edges [Krötzsch et al., 2018]; and reasoning and querying over contextual data, in order to derive conclusions and results valid in a particular setting [Serafini and Homola, 2012, Zimmermann et al., 2012, Schuetz et al., 2021]. In the intersection of data graphs and inductive knowledge, we highlight topics such as similarity-based query relaxation, allowing to find approximate answers to exact queries based on numerical representations (e.g., embeddings) [Wang et al., 2018]; shape induction, in order to learn and formalise inherent patterns in the knowledge graph as constraints [Mihindukulasooriya et al., 2018]; and contextual knowledge graph embeddings that provide numeric representations of nodes and edges that vary with time, place, etc. [Kazemi et al., 2019]. In the intersection of deductive and inductive knowledge, we mention the topics of entailment-aware knowledge graph embeddings [Guo et al., 2016, Demeester et al., 2016], that incorporate rules and/or ontologies when computing plausibility; expressive graph neural networks proven capable of complex classification analogous to expressive ontology languages [Barceló et al., 2020]; as well as further advances on rule and axiom mining, allowing to extract symbolic, deductive representations from the knowledge graphs [Galárraga et al., 2015, Bühmann et al., 2016]. Further challenges arise when considering the creation, enrichment, refinement and publication of knowledge graphs, which call for further works on topics such as automated quality assessment (and repair), distantly-supervised extraction frameworks, efficient access protocols, and anonymisation, to name but a few.

Aside from specific topics, more general challenges for knowledge graphs include scalability, particularly for deductive and inductive reasoning; quality, not only in terms of data, but also the models induced from knowledge graphs; diversity, such as managing contextual or multi-modal data; dynamicity, considering temporal or streaming data; and finally usability, which is key to increasing adoption. Though techniques are continuously being proposed to address these challenges, they are unlikely to ever be completely “solved”; rather they serve as dimensions along which knowledge graphs, and their techniques, tools, etc., will continue to mature.

Given the availability of open knowledge graphs whose quality continues to improve, as well as the growing adoption of enterprise knowledge graphs in various industries, future research on knowledge graphs has the potential to foster key advancements in broad aspects of society. Here we have highlighted just some examples of future research directions of importance to this pursuit.

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