An Algebraic Data Model for Graphs and Hypergraphs (Category Theory meetup, November 2019)
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The document presents an algebraic data model for graphs and hypergraphs, focusing on key concepts from category theory and their application in data modeling. It discusses various data models, including property graphs, and introduces algebraic property graphs that emphasize formal schemas and mapping languages, as well as integration frameworks. Additionally, the document outlines operations on graphs and provides insights into transformations and properties relevant for enterprise data management.
An Algebraic Data Model for Graphs and Hypergraphs (Category Theory meetup, November 2019)
- 1. An Algebraic Data Model for Graphs and Hypergraphs Bonus Intro to Category Theory November 21, 2019 Joshua Shinavier, PhD (Uber)
- 2. • Graphs • Categories • Data Models • Algebraic Property Graphs Outline
- 3. Graphs
- 15. TinkerFactory.createClassic() + vertex labels
- 16. Graph structure
- 17. Categories
- 19. An object
- 22. Composition
- 23. Associativity
- 25. Identity
- 27. An isomorphism
- 28. An isomorphism: f-1 ∘f = ida , f∘f-1 = idb
- 29. An isomorphism
- 32. A category
- 34. A functor
- 36. Graphs and graph operations
- 38. Transformations
- 39. Data models
- 40. • Property Graphs • Resource Description Framework (RDF) • Relational Model • SQL, CQL, HQL, DQL, ... • Data interchange formats • Protocol Buffers, Apache Thrift, Apache Avro • Mostly based on algebraic data types (ADTs) Some data models of interest (@ Uber)
- 41. • A simple interpretation1 I of an RDF graph is: • A non-empty set IR of resources, called the domain or universe of I • A set IP, called the set of properties of I • A mapping IEXT from IP into 𝒫(IR x IR), i.e. the set of sets of pairs <x, y> with x and y in IR • Assuming: • Ground RDF graphs only (⇒ no blank nodes) Resource Description Framework (RDF) 1) World Wide Web Consortium. (2014). RDF 1.1 concepts and abstract syntax.
- 42. • Simple. • Vertices, edges, properties • No standard • W3C Property Graphs working group (2013) • W3C Graph Data Workshop (March 2019) • Next best thing? • TinkerPop’s Graph.Features Property Graph data model
- 43. • Many are concerned with the data model • Vertex/edge id data types • Numeric, string, UUID, “any” • Property support • Supports vertex properties? edge properties? vertex metaproperties? • Multi-properties? • Property data types • Primitive: boolean, byte, double, float, integer, long, string • Complex: array, map, list (uniform/mixed), serializable TinkerPop’s Graph.Features
- 44. Property graph objects v = a vertex label (e.g. Person, Place, Dataset) e = an edge label (e.g. knows, likes, claims) p = a property key (e.g. name, weight) d = a data type (e.g. String, Integer, List<String>)
- 45. Edges
- 47. Edge properties
- 49. Meta-edges: edge to vertex
- 50. Meta-edges: vertex to edge
- 54. A formal data model for enterprise Property Graphs ● Things we need ○ Formal schema & mapping language for property graphs ○ Framework for graph transformation and integration ○ Means to build virtual graphs out of non-graph datasets (and vice versa) ● Algebraic Property Graphs ○ Based on algebraic data types ○ Strong connections to type theory, database theory, and category theory: ■ A familiar approach to type inference ■ Easy to reason about: all operations and proofs mechanically verified in Coq ■ Generalizes to algebraic databases
- 55. Product and sum types, identifiers, and values ● APG: ● Thrift example:
- 56. Algebraic types in common data interchange languages
- 57. Graph, schema, and data
- 58. Natural Transformations of Graphs, Schemas, and Data ● Non-natural transformations are also useful and are described in our paper.
- 59. Taxonomy of graph elements ● Vertex: “no data”. ● Edge: ordered pair of elements. ● Property: ordered pair (element, value). ○ Vertex property: element is a vertex. ○ Edge property: element is an edge ○ Meta-property: element is another property ■ Vertex / edge metaproperty, etc. ● Alias: contains another element ○ Vertex or edge “tags” ○ Aliases for primitive data types ● Hyperelement: everything else
- 60. Useful operations on Graphs, schemas, and data ● Conjunctive and disjunctive queries (select/from/where/union) ○ In: APGs and a predicate. Out: APG ● Views ○ In: APG and APG schema mapping. Out: ‘virtual’ APG ● Convert APGs to algebraic databases, and vice versa ○ In: Schema mapping and APG / algebraic database. Out: algebraic database / APG.