in this post on math.stackexchange, the top voted answer affirmed the following quote:

Roughly speaking, category theory is graph theory with additional structure to represent composition.

I am wondering, are networks categories? s and if so, is category theory ever considered when studying the functioning of various kinds of multi-layer dynamic networks, be they social, biological, etc?

  • 7
    $\begingroup$ No; as the quoted text says, you need "additional structure to represent composition" and networks / graphs don't have this structure. $\endgroup$ Aug 29, 2022 at 21:24
  • $\begingroup$ could we build that additional structure into network models in order to study things that have been traditionally modeled by networks? $\endgroup$
    – neutrino
    Aug 29, 2022 at 23:20
  • 3
    $\begingroup$ You can do anything you want, but as far as things that other people have done, there seems to be more interest in thinking, more or less, about the category of networks than about networks as categories in themselves, see eg arxiv.org/abs/2006.10733. This makes sense, since connections in networks are often simply not composable/transitive: the friend of my friend is not in general my friend, and we want to keep track of the difference! $\endgroup$ Aug 30, 2022 at 1:06
  • $\begingroup$ A graph is certainly a tempting mental image to use in conceptualizing a category, but order to appreciate the necessary structure lacking from a graph, it is helpful to consider graphs with multiple edges between nodes -- the starkest case is of a single node with multiple edges. Given two such self-loops, how is one to compose them? $\endgroup$ Aug 30, 2022 at 2:53
  • $\begingroup$ @KevinArlin how about organizational colleagues for example? i might not be connected to my coworkers coworker directly in terms of projects, but they are my coworker. $\endgroup$
    – neutrino
    Aug 30, 2022 at 5:28


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