MiS Preprint Repository

Delve into the future of research at MiS with our preprint repository. Our scientists are making groundbreaking discoveries and sharing their latest findings before they are published. Explore repository to stay up-to-date on the newest developments and breakthroughs.

MiS Preprint

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.

Jan 6, 2017
Jan 6, 2017

Related publications

2017 Repository Open Access
Emil Saucan and Jürgen Jost

Network topology vs. geometry : from persistent homology to curvature