Efficient Decomposition of Forman-Ricci Curvature on Vietoris-Rips Complexes and Data Applications

Abstract

Discrete Forman-Ricci curvature (FRC) is an efficient tool that characterizes essential geometrical features and associated transitions of real-world networks, extending seamlessly to higher-dimensional computations in simplicial complexes. In this article, we provide two major advancements: First, we give a decomposition for FRC that inhently allows a local computations of FRC. Second, we construct a set-theoretical proof enabling an efficient algorithms for the local computation of FRC in Vietoris-Rips (VR) complexes. Our findings open new avenues for geometric computations in VR complexes and highlight an essential yet under-explored aspect of data analysis and visualisation: the geometry underpinning statistical patterns.