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Network Topology vs. Geometry: From persistent Homology to Curvature
Emil Saucan and Jürgen Jost
We propose our method based on Forman's discretization of Ricci curvature, as an alternative, in the case of Complex Networks, to persistent homology. We show that the proposed method has, among other advantages, the implicity and efficiency of computations. In addition, we gain both expressiveness and computational efficiency by taking into account only those higher dimensional faces that model higher order correlations. In this setting it also has the supplementary advantage of having the capacity of recognizing geometric structures up to homotopy.
We show that the proposed method can be applied also to weighted data, obtained via the geometric, (generalized) Ricci curvature sampling, from manifolds with density. Moreover, we show that the resulting networks can be naturally equipped with the Forman-Ricci curvature, thus representing accurate samplings of the metric, measure and geometric structures of the original weighted manifold.
In addition, we suggest as a method for inferring the real dimension of the data sampled from a geometric object that lacks a manifold structure, the notion of local and statistical dimensions due to Y. Ollivier.