Comparative analysis of Forman-Ricci curvature versions applied to the persistent homology of networks

TL;DR

This paper compares different versions of Forman-Ricci curvature applied to persistent homology in networks, highlighting the effectiveness of triangle-augmented curvature for analyzing non-quasiconvex networks.

Contribution

It extends Forman-Ricci curvature formulas to non-quasiconvex networks and evaluates three versions for persistent homology analysis.

Findings

Triangle-augmented curvature performs best for non-quasiconvex networks.

Plain curvature omits too much information.

Pentagon-augmented curvature distorts results significantly.

Abstract

We provide an overview of Forman-Ricci curvature and persistent homology, and how their combination can be applied to the study of networks. We discuss how the usually employed augmented Forman-Ricci curvature formula, only valid for quasiconvex augmented networks, can be extended to the non-quasiconvex case. We apply three versions of quasiconvex Forman-Ricci curvature (plain, triangle-augmented, and pentagon-augmented) to build time filtrations on non-quasiconvex networks, both model and real-world. Our results suggest that triangle-augmented curvature should be used until the non-quasiconvex formula is further studied, as plain curvature omits too much information, and quasiconvex pentagon-augmented curvature is too rough of an approximation and significantly distorts the results.