The stability of persistent homology of hypergraphs

TL;DR

This paper investigates the stability of persistent embedded homology in hypergraphs, demonstrating its robustness under perturbations and establishing conditions for the invariance of persistent Betti numbers.

Contribution

It proves the stability of persistent embedded homology and homology of associated complexes under filtration perturbations, advancing hypergraph topological analysis.

Findings

Proved stability of persistent embedded homology under filtration perturbations.

Established stability of homology of associated simplicial complexes.

Demonstrated conditions for constancy of persistent Betti numbers.

Abstract

Hypergraph is the most general model for complex networks involving group interactions. Taking the ideas of path homology from Alexander Grigor'yan, Yong Lin, Yuri Muranov and Shing-Tung Yau [18-22], Stephane Bressan, Jingyan Li and the authors of this article introduced embedded homology of hypergraphs [6] in 2019, which has leaded to successful applications in protein-ligand binding network [24, 25] in 2021. A fundamental question arising from practical applications is about the stability of the persistent embedded homology of hypergraphs. In this paper, we prove the stability of the persistent embedded homology as well as the persistent homology of the associated simplicial complex with respect to perturbations of the filtration on a hypergraph. We apply the persistent homology methods to morphisms of hypergraphs and prove the stability with respect to perturbations of the filtrations. We prove the constancy of the persistent Betti numbers under some conditions on the simple-homotopy types of hypergraphs.