Network geometry in complex networks

The spectral dimension, a positive real number related to the probability a random walk on a network will eventually return to where it started, is often finite in geometric networks e.g. the k-nearest neighbour graph on uniformly random points on the torus, so the appearance of finite spectral dimension in a growing network model is often considered to be a “geometric” property. So is the appearance of a non-trivial distribution of node “curvatures” (related to the incidence of triangles/simplicial complexes at a point, see e.g. the recent work of J. Jost, as well as M. Gromov and O. Knill), as well as a non-trivial community structure, much higher clustering than that of random networks, and the famous six degrees of separation i.e. "small world" property. A further example introduced recently is the random topology of the network’s clique complex, where we build a topological space by face-including (gluing together at edges/faces) the many-body interactions e.g. triangles, 4-cliques etc in a network data set, then compare its homology to those of random geometric complexes like the Vietoris-Rips or the Čech complex, introduced by e.g. Linial, Meshulam, Farber, Bianconi and Kahle. We thus introduce and discuss recent progress made on determining to what extent these properties emerge in models of complex networks.