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Knowledge Graphs

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

arXiv:2003.02320v6cs.AIcs.DBcs.LG

TL;DR

Knowledge graphs lack a general unifying account of their models, techniques, uses, and relationship to data management. This tutorial provides a comprehensive introduction covering representation, reasoning, construction, quality, refinement, publication, and applications, while identifying continuing challenges including scalability, quality, diversity, dynamicity, and usability.

  • Problem

    A general unifying summary of how knowledge graphs are used, what techniques they employ, and how they relate to existing data-management topics remains lacking.

  • Method

    The paper synthesizes graph data models, schema, identity, context, deductive and inductive knowledge techniques, graph lifecycle methods, publication practices, and real-world knowledge-graph applications.

  • Results

    The tutorial provides a comprehensive introduction to knowledge graphs, covering their models, representations, reasoning techniques, lifecycle methods, publication principles, and adoption in practice.

  • Takeaways & Limitations

    Knowledge graphs bring together techniques for integrating and extracting value from diverse data, with future development centered on scalability, quality, diversity, dynamicity, and usability.

  • Takeaways & Limitations

    Knowledge-graph challenges such as scalability, quality, diversity, dynamicity, and usability are ongoing dimensions unlikely to be completely solved.

Abstract

from arXiv · show

In this paper we provide a comprehensive introduction to knowledge graphs, which have recently garnered significant attention from both industry and academia in scenarios that require exploiting diverse, dynamic, large-scale collections of data. After some opening remarks, we motivate and contrast various graph-based data models and query languages that are used for knowledge graphs. We discuss the roles of schema, identity, and context in knowledge graphs. We explain how knowledge can be represented and extracted using a combination of deductive and inductive techniques. We summarise methods for the creation, enrichment, quality assessment, refinement, and publication of knowledge graphs. We provide an overview of prominent open knowledge graphs and enterprise knowledge graphs, their applications, and how they use the aforementioned techniques. We conclude with high-level future research directions for knowledge graphs.

1 INTRODUCTION

Knowledge graphs use graph-based models to represent and connect knowledge from diverse data sources, supporting flexible evolution and large-scale integration. This tutorial addresses the field’s fragmented literature with a comprehensive introduction spanning representations, reasoning, construction, assessment, refinement, publication, and applications.

  • Motivation: Knowledge graphs represent data as graphs whose edges capture relations among entities, supporting integration and value extraction from diverse large-scale sources.Graph abstractions can also postpone schema definition, allowing scope and data to evolve flexibly and accommodating incomplete knowledge.
  • Scope: The tutorial covers graph data models and query languages, schema, identity and context, deductive and inductive techniques, graph creation and enrichment, quality, refinement, publication, and applications.Its intended audience is researchers and practitioners new to knowledge graphs, without assumed specialist expertise.
  • Definition: The paper adopts an inclusive definition of knowledge graphs as graph-structured data intended to accumulate and convey knowledge about the real world.Nodes represent entities of interest, while edges represent relations between those entities.
  • Representations and knowledge: Knowledge graphs require schema, identity, and context representations to manage diversity in structure, granularity, and references to real-world entities.The paper also discusses deductive and inductive techniques for representing, entailing, extracting, and accumulating additional knowledge.
  • Related literature: Existing tertiary literature tends to focus on specific knowledge-graph aspects, whereas this tutorial aims to provide a broad and accessible introduction.The paper identifies the absence of a general unifying summary connecting knowledge-graph uses, techniques, and data-management topics.

2 DATA GRAPHS

Knowledge graphs begin with graph abstractions that represent entities and relations, offering flexible alternatives to upfront relational schemas. The section contrasts directed edge-labelled, heterogeneous, RDF, and property graph models, including their trade-offs and applications.

  • Graph foundations: Relational modelling becomes costly when event data evolve to include multiple names, venues, types, and incomplete dates.The tourism example decomposes one table into relations that support 0–n names, venues, and types and 0–1 start and end dates.
  • Graph foundations: Graph models can defer upfront schema design, enabling new binary relations and more flexible integration of diverse sources.Graphs also represent and query cycles without imposing a hierarchy on the data.
  • Graph models: Directed edge-labelled graphs provide a minimal representation, while heterogeneous graphs assign types to nodes and edges as part of the model.RDF standardises directed edge-labelled graphs with IRIs, literals, and blank nodes for identifying entities and representing values.
  • Graph foundations: Graph data models represent entities as nodes and relations as edges, forming the initial data graph underlying a knowledge graph.
  • Graph models: Property graphs attach labels and property–value pairs to nodes and edges, concisely modelling contextual relations such as companies offering flights.They can be translated to and from directed edge-labelled graphs without information loss, so implementation factors often guide model choice.

3 SCHEMA, IDENTITY, CONTEXT

Knowledge graphs extend data graphs with representations of schema, identity, context, ontologies, and rules. This section distinguishes semantic and validating schemata, explains open-world assumptions, and illustrates shape-based validation and conformance.

  • Knowledge graphs are data graphs potentially enhanced with schema, identity, context, ontologies, and rules, either embedded in or layered above the graph.
  • Graph schemata can be semantic, validating, or emergent, while allowing graph models to postpone schema definition when flexibility is needed.
  • Semantic schemata: Semantic schemata define meanings and relationships among classes and properties, including subclasses, subproperties, domains, and ranges, enabling inference.RDFS is a prominent standard for expressing these semantic features in RDF graphs.
  • Identity and context: The open-world assumption treats missing edges as unknown, whereas local or closed-world assumptions can support completeness requirements for selected data.The paper presents the open-world assumption as the appropriate default for diverse, incomplete, large-scale graph data.
  • Validating schemata: Validating schemata specify constraints on existing graph data, such as required event properties and expected node types, and report violations.Unlike semantic schemata, they validate existing data rather than infer new graph data.
  • Shapes and conformance: Shapes target nodes and impose constraints whose satisfaction determines conformance, including inherited, recursive, and inter-shape requirements.Examples show that missing required properties cause nonconformance, while open shapes permit additional unspecified properties.

4 DEDUCTIVE KNOWLEDGE

Deductive knowledge uses formal semantics, ontologies, and rules to derive knowledge not explicitly present in a graph. The paper explains how these mechanisms support entailment while exposing semantic and computational limits.

  • Ontologies: Ontologies provide graph-representable formal knowledge structures for expressing and automating complex entailments.They connect term meanings, logical frameworks, and reasoning procedures.
  • Ontologies: An ontology’s usefulness depends on agreement about its definitions, detail, and adoption, with broader agreement enhancing interoperability across knowledge graphs.
  • Semantics: If-and-only-if semantics can entail more ontology-language axioms than if–then semantics, and OWL generally uses the former.
  • Rules: Rules encode entailments under ontological conditions, but rule sets such as OWL 2 RL/RDF remain incomplete for negation, existential, universal, and counting features.
  • Entailment: Entailment is directional: one graph may have a model that the other lacks, while the converse can also hold, so neither graph entails the other.

5 INDUCTIVE KNOWLEDGE

Inductive knowledge generalises patterns from observations to produce potentially imprecise predictions, using analytics, embeddings, neural models, and symbolic learning techniques. These methods support graph analysis and knowledge extraction but can be sensitive to graph representation choices.

  • 5.1 Inductive knowledge: Inductive knowledge generalises patterns from input observations to generate novel predictions that may be associated with confidence rather than certainty.The paper illustrates this with capital cities and international airports, where the observed pattern holds in 187 of 195 cases, giving confidence 0.959.
  • 5.1 Graph analytics: Graph analytics discovers structural patterns such as central nodes, communities, connectivity vulnerabilities, node similarity, and paths.Examples include identifying transport hubs, tourist-attraction groupings, brittle routes, similar attractions, and routes between pairs of nodes.
  • 5.2 Neural and embedding models: Knowledge graph embeddings use self-supervision to learn low-dimensional numeric models, while neural models process graph features and recursively updated node states.Tensor decomposition and convolutional approaches provide alternative embedding architectures; TuckER is reported as achieving state-of-the-art results on standard benchmarks.
  • 5.1.5 Analytics with entailment: Analytics results can depend substantially on whether edge types are projected, weighted, or retained, especially when methods account for edge direction.The paper identifies semantically invariant analytics—yielding the same results over semantically equivalent graphs—as an open research direction.
  • 5.4 Symbolic learning: Symbolic learning includes rule mining and axiom mining, which derive quantified logical patterns that can be applied to unseen examples.Rule-mining quality is assessed using support and confidence, representing the number and ratio of positive entailments, respectively.

6 CREATION AND ENRICHMENT

Knowledge graphs can be created and enriched from human contributions, text, markup, and existing knowledge graphs. Text extraction commonly combines entity and relation processing with alignment and joint task frameworks.

  • Creation methodology depends on the actors, domain, applications, and available sources, while graph flexibility supports incremental construction.
  • Human editors contribute knowledge through in-house work, crowdsourcing, feedback, and collaborative platforms, but this involvement incurs high costs.
  • Text-based construction applies NLP and information extraction because achieving high precision and recall from rich corpora is non-trivial.
  • The main text-extraction tasks identify entities, link mentions to graph nodes, disambiguate contexts, and extract binary or n-ary relations.
  • Open relation extraction may require aligning extracted relations with graph predicates using mappings, rules, similarity methods, clustering, or linguistic techniques.
  • Frameworks may jointly perform tasks such as word-sense disambiguation and entity linking or named-entity recognition and relation extraction to improve multiple tasks together.

7 QUALITY ASSESSMENT

Knowledge graph quality is assessed as fitness for purpose across multiple dimensions, because graphs built from external sources may be incomplete, duplicated, contradictory, or incorrect. The section defines dimensions and corresponding ways to measure them.

  • Quality assessment is necessary because initial knowledge graphs commonly contain incomplete, duplicate, contradictory, or incorrect statements from multiple sources.
  • Accuracy concerns whether nodes and edges represent real-world phenomena correctly, including syntactic accuracy, semantic accuracy, and timeliness.
  • Syntactic accuracy can be measured as the ratio of incorrect values for a property to its total values, often using validation tools.
  • Completeness covers schema, property, and population aspects, whereas coverage concerns omission of domain-relevant elements.
  • Bias assessment considers geographic, linguistic, social, and schema biases that may under-represent entities, relations, languages, or uncommon cases.
  • Coherency includes freedom from logical contradictions and constraint violations, measurable by counting inconsistencies or violations per constraint.
  • Succinctness evaluates whether graph content is relevant, concise, and intelligible, including intensional and extensional conciseness.

8 REFINEMENT

Knowledge graph refinement improves an existing graph by completing missing information and correcting existing errors. Completion predicts plausible absent edges, while correction validates or repairs present edges.

  • Refinement differs from creation and enrichment because it generally does not ingest content from external sources through extraction or mapping.
  • Knowledge graph completion fills missing edges deemed correct but neither given nor entailed, often using link prediction techniques.
  • Link prediction addresses general, type, and identity links, with type prediction reducible to classification and identity prediction analogous to entity matching.
  • Identity matching combines value matchers for property similarity with context matchers based on surrounding nodes and edges.
  • Pairwise matching requires O(n^2) comparisons, so blocking groups similar entities before matching to improve efficiency.
  • Correction removes incorrect existing edges through fact validation, which scores plausibility, or inconsistency repair, which resolves violations using ontological axioms.
  • Fact-finding methods can represent facts and sources in multipartite graphs and jointly estimate source trustworthiness and fact plausibility.

9 PUBLICATION

Publishing knowledge graphs makes all or part of their content accessible, often on the Web, while principles and protocols guide discoverability, access, interoperability, reuse, and linking. FAIR emphasizes reusable machine-readable data, and Linked Data provides a technical Web-based approach.

  • Publication may be desirable or required for volunteer, publicly funded, or governmental knowledge graphs, but not for enterprise graphs offering competitive advantage.
  • Publication guidance covers FAIR and Linked Data principles, access protocols, and techniques for restricting access or usage when appropriate.
  • FAIR comprises findability, accessibility, interoperability, and reusability goals applicable to data, metadata, or both.
  • FAIR findability uses persistent identifiers, rich metadata, explicit metadata-to-data identifiers, and searchable registration.
  • FAIR interoperability requires accessible shared knowledge-representation formalisms, while reusability includes licensing, provenance, and domain-relevant standards.
  • Linked Data documents distinguish IRIs for entities from IRIs for documents, allowing linked graphs to provide additional descriptions through dereferencing.
  • Open PHACTS and UniProt are cited as FAIR knowledge graphs that provide graph views through RDF.
  • Linked Data uses IRIs, HTTP lookup, useful returned content, and links to related entities to interconnect RDF graphs into a Web of Data.

10 KNOWLEDGE GRAPHS IN PRACTICE

Knowledge graphs have been deployed as open and enterprise resources across domains, using different sources, schemas, and curation models. Their applications range from unified data access and multilingual knowledge to search, recommendations, social networking, and analytics.

  • Open knowledge graphs: Open knowledge graphs are publicly accessible resources published under the Open Data philosophy, often as Linked Open Datasets following Linked Data principles.Linked Open Datasets are RDF graphs published under the Open Data philosophy.
  • Open knowledge graphs: DBpedia extracts structured knowledge from Wikipedia and links it to external resources, while supporting multiple schemas for different application requirements.Its linked resources include datasets such as DailyMed, DrugBank, GeoNames, MusicBrainz, New York Times, and WordNet; supported schemas include SKOS, YAGO, UMBEL, and the DBpedia ontology.
  • Open knowledge graphs: YAGO combines Wikipedia extraction with WordNet’s hierarchy to provide a lightweight ontology intended for tasks including machine translation, query expansion, classification, cleaning, and integration.The passage characterizes YAGO as having high quality and coverage.
  • Open knowledge graphs: Freebase addressed large-scale integration by soliciting contributions from human editors rather than relying primarily on extraction from Wikipedia and WordNet.Its motivation included challenges from the decentralized Semantic Web, including uneven adoption, implementation difficulty, and distributed query performance.
  • Open knowledge graphs: Wikidata centralizes collaboratively edited knowledge to reduce inconsistencies from manually maintaining related facts across Wikipedia articles and language versions.A fact added once can trigger automatic updates for many clients and articles.
  • Enterprise knowledge graphs: Enterprise knowledge graphs support applications such as search, recommendations, conversational agents, advertising, analytics, social connections, and multilingual services.Google emphasizes identifying entities rather than matching only strings, while Amazon applies its graph to product search and recommendations.

11 SUMMARY AND CONCLUSION

The paper synthesizes knowledge graphs as graph-based systems for accumulating and conveying real-world knowledge, covering their models, representations, techniques, quality practices, publication, and real-world adoption. It identifies future research as an intersection of multiple disciplines shaped by persistent challenges in scale, quality, diversity, dynamicity, and usability.

  • Summary and conclusion: The paper’s comprehensive introduction covers graph data models, schema, identity, context, deductive and inductive techniques, creation, enrichment, quality, refinement, publication, and adoption.Its adopted definition treats a knowledge graph as data structured with entity nodes and relational edges to convey real-world knowledge.
  • Future directions: Future research can combine graph databases, knowledge representation, logic, machine learning, graph algorithms, ontology engineering, data quality, language processing, information extraction, privacy, and security.The paper presents this convergence as a way to maximize knowledge and value distilled from diverse large-scale sources through graph-based abstraction.
  • Future directions: Intersections between data graphs and deductive knowledge motivate work on formal semantics for property graphs and reasoning over contextual data.These directions concern languages that account for labels and property–value meanings and systems that derive conclusions from contextual information.
  • Future directions: Persistent challenges include scalability, quality, diversity, dynamicity, and usability, spanning both knowledge-graph data and models induced from those graphs.The paper frames these as continuing dimensions of maturation rather than challenges likely to be completely solved.
  • Conclusion: Improving open-graph quality and expanding enterprise adoption give knowledge-graph research potential to advance broad aspects of society.The conclusion presents this as a potential enabled by improving open resources and growing industrial use.

A BACKGROUND

The background section situates the paper within the historical development of knowledge graphs and prior efforts to define the concept. It emphasizes that the review aims to be comprehensive while building on, rather than replacing, earlier discussions.

  • Background: The section reviews the broader historical context leading to modern knowledge graphs and definitions proposed before and after Google’s 2012 announcement.It treats the Google Knowledge Graph announcement as a reference point in the modern history of the term.
  • Background: The discussion builds on prior treatments by Ehrlinger and Wöß and by Bergman without claiming to subsume them.The cited works are referred to for further details.
  • Background: The section states that the paper’s goal is comprehensive coverage of the background and definitions of knowledge graphs.This frames the historical review as contextual support for the paper’s broader introduction.

A.1 Historical Perspective

The historical perspective traces knowledge graphs to diagrammatic and computational traditions for representing and reasoning about knowledge. It connects these traditions to modern standards and open knowledge graphs shaped by the Semantic Web.

  • Historical perspective: Knowledge representation through diagrams developed from early visual reasoning traditions, including Aristotle’s work, Euler circles, Venn diagrams, and later formal systems by Sylvester, Peirce, and Frege.These systems supported visual reasoning and, in some cases, codified reasoning.
  • Historical perspective: Digital computers enabled programs to perform formal reasoning and encode knowledge representations, leading through semantic networks and related systems toward modern knowledge-representation approaches.The passage identifies work by Ritchens, Quillian, Travers, and Milgram as part of this development.
  • Historical perspective: Semantic networks lacked well-defined semantics, whereas predicate calculus offered precise model-theoretic semantics but was less suited to a messy, incompletely explored real world.The contrast motivates the continuing need to balance formal precision with open-world complexity.
  • Historical perspective: The historical tradition established graph-based representation, computational reasoning, combinations of logical and statistical reasoning, and multimodal data as sources of knowledge.These notions remain relevant to knowledge graphs.
  • Historical perspective: Conceptual graphs, semantic networks, and frames preceded Description Logics, which sought well-defined semantics for practical reasoning systems with decidable logics.These developments helped underpin later standards and techniques including RDF, OWL, RDFS, SPARQL, Linked Data, and Shape Expressions.
  • Historical perspective: Many prominent open knowledge graphs, including BabelNet, DBpedia, Freebase, Wikidata, and YAGO, emerged from or later adopted Semantic Web standards.The passage links these resources to the standards tradition discussed in the paper.

A.2 “Knowledge Graphs”: Pre 2012

Before 2012, “knowledge graph” referred to multiple independently developed graph-based representations across education, medicine, linguistics, planning, organisations, and text analysis. These works used varied node, edge, weighting, and dependency semantics, often on relatively small graphs.

  • A.2 “Knowledge Graphs”: Pre 2012: Pre-2012 authors independently used “knowledge graph” for different applications and meanings.Examples include instructional courses, medical expert systems, linguistic representations, organisational knowledge, and explanation knowledge.
  • A.2 “Knowledge Graphs”: Pre 2012: Pre-2012 systems included weighted graphs refined by learning and graphs extracted from examples or text.These approaches supported gradual weight adjustment, symbolic learning, and extraction of causal or explanatory knowledge.
  • A.2 “Knowledge Graphs”: Pre 2012: Earlier knowledge graphs commonly represented dependencies, causality, conjunctions, disjunctions, or graded belief through edges.Some works used weighted edges, while others used and–or structures to encode conjunctive or disjunctive dependencies.
  • A.2 “Knowledge Graphs”: Pre 2012: Knowledge graphs from 1970–2000 were generally small compared with modern graphs containing millions or billions of nodes.The passage attributes the earlier scale difference to more limited computational resources and fewer available structured-data sources.

A.3 “Knowledge Graphs”: 2012 Onwards

From 2012 onward, Google’s announcement popularised “knowledge graph” in research and industry, increasing the need for precise definitions. Academic definitions range from simple graph structures to graph-structured knowledge bases and example-based characterisations.

  • A.3 “Knowledge Graphs”: 2012 Onwards: Google announced its Knowledge Graph in 2012, after which mentions of knowledge graphs rapidly increased in research literature.The announcement described a graph representing real-world entities and their relationships and targeted a broad audience.
  • A.3 “Knowledge Graphs”: 2012 Onwards: Growing academic attention made formal definitions necessary for characterising knowledge graphs, their structure, and their uses.The paper groups prior definitions into four general categories.
  • A.3 “Knowledge Graphs”: 2012 Onwards: Category I defines a knowledge graph as a graph whose nodes represent entities and whose edges represent relationships.This formulation is often expressed as a directed edge-labelled graph, binary relations, or triples.
  • A.3 “Knowledge Graphs”: 2012 Onwards: Category II treats a knowledge graph as a graph-structured knowledge base, raising questions about how knowledge bases differ from ontologies.Some authors distinguish knowledge graphs from ontologies and criticise treating Google’s graph as a knowledge base.
  • A.3 “Knowledge Graphs”: 2012 Onwards: Technical definitions may combine extensional graph data with intensional inference rules and derived data.This view extends a knowledge base with reasoning over its represented constructs.
  • A.3 “Knowledge Graphs”: 2012 Onwards: Category IV characterises knowledge graphs by examples such as DBpedia, Google’s Knowledge Graph, Freebase, and YAGO.The paper notes that this approach defines the category extensionally rather than resolving the definitional issue.

B FORMAL DEFINITIONS

The appendix formalises graph data models, datasets, and query constructs used throughout the paper. It defines directed edge-labelled, heterogeneous, and property graphs, then extends them with datasets, graph patterns, and navigational path expressions.

  • B FORMAL DEFINITIONS: The formal appendix uses example-driven explanations alongside formal definitions to make foundational concepts accessible.It abstracts away from several model-specific issues, including blank-node semantics and literal entailment.
  • B.1 Data Graph Models: A directed edge-labelled graph consists of nodes, edge labels, and labelled directed edges represented as triples.The definition permits overlap between nodes and labels and allows isolated nodes or labels.
  • B.1.2 Heterogeneous graph: Heterogeneous graphs add labels for nodes and edges, allowing multiple edge types between the same pair of nodes.The examples distinguish node types such as City and Country from edge types such as capital and country.
  • B.1.3 Property graph: Property graphs represent node and edge identifiers, labels, and property–value pairs through explicit mappings.The model permits overlapping sets and multiple values for a property, treating such variations as syntactic.
  • B.1.4 Graph dataset: A graph dataset contains a default graph and optionally a uniquely named set of additional graphs.An RDF dataset is a standardised graph dataset whose graphs are RDF graphs and whose names may be blank nodes or IRIs.
  • B.2 Querying: Graph querying is formalised using graph patterns containing constants and variables in node and edge positions.The treatment introduces terms as the union of constants and variables and defines triple-pattern structures.
  • B.2.3 Navigational graph patterns: Path expressions support inverse, repetition, concatenation, and disjunction, with evaluation returning endpoint pairs.Concatenation joins paths through an intermediate node, and property-graph evaluation adapts the base edge-label case.
  • B.2.3 Navigational graph patterns: Path-expression semantics for property graphs are defined analogously, after which the remaining navigational definitions are unchanged.The property-graph case evaluates labelled edges through the edge and label mappings.

B.6.1 Graph parallel frameworks.

Graph parallel frameworks compute node representations by exchanging messages and aggregating neighbourhood information, while embeddings map entities and relations into numeric spaces for plausibility scoring. The appendix illustrates these abstractions with PageRank and summarises expressivity trade-offs.

  • B.6.1 Graph parallel frameworks: A directed vector-labelled graph supplies nodes, edges, and feature vectors as the input state for graph-parallel computation.Node and edge types or attributes can be encoded in fixed-dimensional feature vectors.
  • B.6.1 Graph parallel frameworks: The PageRank example initializes node vectors with a score, outdegree, and graph size, then propagates damped scores.A fixed number of iterations provides one possible termination condition.
  • B.6.1 Graph parallel frameworks: A graph parallel framework consists of message, aggregation, and termination functions.Messages pass along edges, aggregation updates node vectors, and termination determines when computation stops.
  • B.6.1 Graph parallel frameworks: The framework’s default formulation dynamically computes node vectors while passing messages to outgoing neighbours.The definitions can be adapted to dynamic edge vectors or incoming-neighbour message passing.
  • B.6.1 Graph parallel frameworks: Graph-parallel computation primarily requires local neighbourhood information at each step, though systems may also support global operations.Possible extensions include dynamically modifying the graph and sharing global computation results.
  • B.6.2 Knowledge graph embeddings: Knowledge graph embeddings map nodes and edge labels to vectors or matrices and score triples using a plausibility function.Higher scores indicate that an edge is considered more plausible.
  • B.6.2 Knowledge graph embeddings: Embedding models trade parameter count and computational cost against expressiveness.Some models are formally fully expressive given sufficient dimensionality, although the required dimensions may be impractical.

A property graph

A property graph is modeled with nodes, edges, labels, properties, and values, while graph patterns specify structures to match against data graphs. Their evaluation returns variable mappings satisfying the pattern, with complex patterns built recursively from projection, selection, and combination operators.

  • A property graph pattern is a tuple containing node and edge identifiers, labels, properties, values, and mappings for connectivity, labels, and property–value pairs.
  • Complex graph patterns: The framework defines partial variable-to-constant mappings, projection, selection satisfaction, and compatibility to support complex-pattern evaluation.
  • Sub-graph containment requires inclusion of nodes, edges, and labels, while property-graph containment additionally accounts for the corresponding property-graph structure.
  • Graph-pattern evaluation returns mappings whose instantiated pattern is a sub-graph of the data graph and whose domain covers all variables in the pattern.
  • The paper adopts homomorphism-based semantics, whereas isomorphism-based semantics can exclude mappings that identify distinct node or edge variables with the same constant.
  • Complex graph patterns: Complex graph patterns are defined recursively using graph patterns, projection, boolean selection conditions, and operators such as join, union, difference, and left-join.
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