The subject of network geometry (Boguna et al., Network Geometry, Nature Reviews Physics 2021) looks at "geometric aspects" of complex networks.

This is about studying a metric on the nodes, though can involve diffusion geometry, and fractal geometry as well.

Can one work with an "Erlangen program" for a complex network? Though each node is not the same given the high heterogeneity of the network (they are not in a "homogeneous space"), there is some "statistical symmetry", given the complex networks can be naturally embedded in hyperbolic space (D. Krioukov, Clustering Implies Geometry in Networks, Phys. Rev. Lett. 2016).

I am basically trying to understand if network geometry, in this sense, can be thought of in terms of group theory and projective geometry.

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    $\begingroup$ Graphs, geometry and symmetry interact very strongly in the area of geometric group theory, usually using the natural path metric on the graph. I don't know if that fits into the framework of "network geometry". $\endgroup$
    – HJRW
    Jun 10 at 21:11
  • $\begingroup$ Ok very interesting. So the automorphism group of the graph would then have a geometric structure? $\endgroup$
    – apkg
    Jun 11 at 17:45
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    $\begingroup$ Not usually the full automorphism group; geometric group theory is primarily concerned with discrete groups acting geometrically on graphs. But there are some strands where the discrete group is thought of as a lattice in the full automorphism group. It’s a big subject, so hard to summarise, but if you want an overview you could do worse than look at the recent book “Office hours with a geometric group theorist”. $\endgroup$
    – HJRW
    Jun 12 at 9:50

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