The document discusses algebraic property graphs (APGs) as a mathematical framework that connects graph and non-graph data models, addressing the limitations of existing implementations. It emphasizes the theoretical foundations rooted in category theory and proposes APGs for enhancing data representation strategies. Additionally, it includes references to implementations and further research related to the concept.

1. Algebraic Property Graphs
Joshua Shinavier
Ryan Wisnesky
https://arxiv.org/pdf/1909.04881.pdf
DICE colloquium presentation
Adrian Wilke
2019-10-25
https://dice-research.org/

2. Authors
2
Joshua Shinavier
Uber Technologies
One of the three co-founders of TinkerPop [JS]
image: https://on.mktw.net/2MbIZTW
Ryan Wisnesky
Conexus AI
“[...] algebraic property graphs as a simple but
mathematically rigorous bridge between graph and
non-graph data models [...]”
B.S. and M.S. degrees in mathematics and computer
science from Stanford University and a Ph.D. in
computer science from Harvard University, where he
studied the design and implementation of provably
correct software systems. Previously, he was a
postdoctoral associate in the MIT department of
mathematics, where he developed the categorical
query language, CQL. He currently leads open-source
and commercial development of CQL as CTO of
Conexus AI.
https://www.wisnesky.net
APG listed as case study for Categorical Query
Language (CQL) at categoricaldata.net/papers

3. Contents
3
15 text pages
7.6 pages appendix
2.3 pages toc & ref

4. Contents
4
“main contribution”
motivation
short
introduction

5. Motivation: Property graphs ⇄ RDF
5
slide source: [CDL]

6. Motivation: Property graphs ⇄ RDF
6
slide source: [CDL]

7. Motivation: Property graphs
7
also: missing semantics in several implementations
slide source: [CDL]

8. Motivation: Hypergraphs
8
slide source: [CDL]

9. Category theory
9

10. Category theory
10
“In mathematics, abstract nonsense [...] are terms
used by mathematicians
to describe abstract methods related to
category theory.”
https://en.wikipedia.org/w/index.php?title=Abstract_nonsense&oldid=881291659

11. Category theory
11

12. Category theory: Morphisms and composition
12

13. example 2 source: [EC]
Category theory: Morphisms and composition
13

14. Category theory: Functors
14

15. Algebraic Property Graphs
Definition
15

16. AGP:
- P
- t ∈ P primitive types
- V(t) primitive values
- four sets
- four functions
- equation
16
AGP: definition

17. 17
AGP: definition
labels: think of
classes in OOP

18. 18
AGP: definition
t1
⨉ t2
e.g. for edges
edge(nodeA, nodeB)
think of types in OOP

19. 19
AGP: definition
value used for vertex
labels
elements: think of
objects in OOP

20. AGP: definition
20
elements: think of
objects in OOP

21. AGP: definition
21
labels: think of
classes in OOP
lambda like label
elements: think of
objects in OOP

22. AGP: definition
22
sigma 🠖 schema
tau 🠖 type G(𝓣): types of G
G(𝓣): types of G

23. AGP: definition
23

24. AGP: definition
24
Composition
g ○ f
g( f(x) )
Image: https://en.wikipedia.org/wiki/Morphism

25. Taxonomy
&
examples
25

26. Taxonomy: Vertices
26

27. Taxonomy: Vertices
27

28. Taxonomy: Edges
28
t1
⨉ t2
e.g. for edges
edge(nodeA, nodeB)
We already saw this.

29. Taxonomy: Edges
29

30. Taxonomy: Properties
30

31. Taxonomy: Properties
31

32. Migrating AGPs
32

33. Migrating AGPs
33
obviously not.

34. RDF ⇄ AGP
34
free category
is the category that results from freely concatenating arrows together,
whenever the target of one arrow is the source of the next.

35. RDF ⇄ AGP
35

36. Summary
36
“We propose algebraic property graphs as a
simple but mathematically rigorous
bridge between graph and non-graph data models.”

37. Summary
37
“main contribution”
motivation
short
introduction

38. For further research
38
● Java implementation
https://github.com/CategoricalData/CQL/tree/master/src/main/java/catdata/apg
● Coq proofs (paper footnote 9) https://www.categoricaldata.net/APG.v
● [CDL] Joshua Shinavier: In Search of the Universal Data Model (Connected Data London 2019)
https://youtu.be/telyBQCuq70
● Joshua Shinavier: Algebraic Property Graphs (GQL Community Update)
https://www.slideshare.net/joshsh/algebraic-property-graphs-gql-community-update-oct-9-2019
● Research on CQL https://www.categoricaldata.net/papers
Related to RDF: “Formal Modelling and Application of Graph Transformations in the Resource
Description Framework” (doctoral dissertation)
● [JS] Joshua Shinavier: Position statement for the W3C Workshop on Graph Data, 2019
https://www.w3.org/Data/events/data-ws-2019/assets/position/Joshua%20Shinavier.pdf
● [EC] Category Theory in Life - Eugenia Cheng https://youtu.be/ho7oagHeqNc