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Stream: deprecated: mathematics

Topic: Knowlege graph fusion as a global section in a presheave

Caleb Foong (May 26 2021 at 19:20):

I am working on a problem to fuse different knowledge graphs together. The graph might share some entities, but the mapping might not be unique. I am thinking about the knowledge graph as a presheave and if there is a coherent way to combine all of the graphs then there should be global section. Any idea if this will work or any pointer to other resources will be helpful. Thanks

Henry Story (May 26 2021 at 19:45):

Btw. what is the mathematical definition you are using for "knowledge graphs"? It seems to be widely used in some parts of the semantic web community...

Caleb Foong (May 27 2021 at 02:27):

I think the "knowledge graph" can be represented by a hypergraph + some inference rules. For the purpose of fusion, I will regard them as just hypergraphs.

Matteo Capucci (he/him) (May 27 2021 at 09:13):

I assume you are regarding knowledge graphs as presheaves on small categories. Since the shape of the graph is not fixed, we are looking at the category $\int^{C \in \mathbf{Cat}} \mathbf{Psh}(C)$ (the category of elements of small presheaves), or more explicitly at the category of pairs $(C, G)$ of a base category $C$ and a presheaf $G : C^{op} \to \mathbf{Set}$ on it, whose morphisms are themselves pairs $(F, F^\sharp) : (C,G) \to (D, H)$ of a functor $F:C \to D$ and a natural transformation $F^\sharp : G \to F^*H$ (here $F^*H = H \circ F$).
Then fusion should amount to taking the colimit of a diagram in this category, a diagram describing the glueing procedure.
Global sections of the colimit should admit an explicit description, which I can't get off the top of my head now.

Caleb Foong (May 27 2021 at 17:06):

in particular I am looking at TypeDB:
https://grakn.ai/
if you look at their way to defining the schema, I would say it is more like simplicial complex rather than an ordinary category.