## Higher Dimensional Models of Networks

David Spivak's 2009 article Higher-dimensional models of networks gives a very helpful overview of the relation between graphs, hypergraphs and simplicial sets, as different ways of describing networks, or indeed as generalizations of the notion of graphs themselves. [1]

One intriguing claim by David Spivak is that simple graphs are not powerful enough to model certain types of networks, especially networks involving group communication. Here simplicial sets make their appearance (see this PBS video intro Your Brain as Math to get a quick taste). A similar point is made in a 2020 paper Networks beyond pairwise interactions: Structure and dynamics which takes detailed examples from many sciences. Still, when I heard of a similar argument many years ago, I could not see how this could be quite true. For since RDF is a first order logic (see Bicategories of Relations article [1]), then how could it not be able to describe group communication? Indeed we have been describing groups since 2004 with the friend of a friend ontology and have been building decentralised social networks with those tools! The only way I could see that making sense is if these groups involves something higher dimensional than first order logic, which would point to modal logic.

And indeed looking in the direction of modal logic I found the very clearly argued paper from Feb 2020 Knowledge and simplicial complexes which shows how multi agent epistemic and doxastic (dynamic) modal logic can be modeled with simplicial complexes.

What do we have in RDF that would allow one to build higher dimensional networks? Linked Data and RDF Quads (a.k.a RDF Datasets) seem to fit the bill. (see my short presentation at SemWeb Pro). It is not the same thing if one RDF document contains the two statements

```
@prefix foaf: <http://xmlns.com/foaf/0.1/> .
<#me> foaf:knows <https://www.w3.org/People/Berners-Lee/card#i> .
<https://www.w3.org/People/Berners-Lee/card#i> foaf:knows <#me> .
```

which would be a self-assertion of knowledge, and the publication of those two triples in different documents, one at https://bblfish.net/people/henry/card

`<#me> foaf:knows <https://www.w3.org/People/Berners-Lee/card#i> . `

and the other at https://www.w3.org/People/Berners-Lee/card

`<#i> foaf:knows <https://bblfish.net/people/henry/card#me> . `

which one can think of as two speech acts, or rather document-acts. This smacks of a higher dimensional network. But how should one model it? Perhaps Simplicial Sets together with RDF can be helpful here?

[1] Missing from Spivak's analysis on higher dimensional networks are I think globular sets which allow one to model graphs with arrows between arrows, and which can be used as finite presentations of n-categories). So for example I think that the 3 hypergraph structure of RDF `s,r,o: A \to N`

, gives rise to 2-graphs since an arrow `a \in A`

can have a relation `r(a)`

which is the subject or object of another arrow. This should explain how the triple structure of RDF turns out to have a categorical interpretation in the bicategories of relations.