Efficient set-theoretic algorithms for computing high-order Forman-Ricci curvature on abstract simplicial complexes

TL;DR

This paper introduces an efficient set-theoretic algorithm for computing high-order Forman-Ricci curvature on simplicial complexes, significantly reducing computational costs and enabling advanced topological data analysis.

Contribution

The authors develop a novel set-theoretic formulation and software implementation, FastForman, to efficiently compute high-order FRC, overcoming previous computational limitations.

Findings

Set-theoretic formulation accelerates FRC computation

FastForman outperforms existing implementations in benchmarks

Enables analysis of high-dimensional complex data sets

Abstract

Forman-Ricci curvature (FRC) is a potent and powerful tool for analysing empirical networks, as the distribution of the curvature values can identify structural information that is not readily detected by other geometrical methods. Crucially, FRC captures higher-order structural information of clique complexes of a graph or Vietoris-Rips complexes, which is not readily accessible to alternative methods. However, existing FRC platforms are prohibitively computationally expensive. Therefore, herein we develop an efficient set-theoretic formulation for computing such high-order FRC in simplicial complexes. Significantly, our set theory representation reveals previous computational bottlenecks and also accelerates the computation of FRC. Finally, We provide a pseudo-code, a software implementation coined FastForman, as well as a benchmark comparison with alternative implementations. We envisage that FastForman will be used in Topological and Geometrical Data analysis for high-dimensional complex data sets. Moreover, our development paves the way for future generalisations towards efficient computations of FRC on cell complexes.