Differential Calculus on Hypergraphs and Mayer-Vietoris Sequences for the Constrained Persistent Homology

Abstract

In this paper, we study the discrete differential calculus on hypergraphs by using the Kouzul complexes. We define the constrained (co)homology for hypergraphs and give the corresponding Mayer-Vietoris sequences. We prove the functoriality of the Mayer-Vietoris sequences for the constrained homology and the functoriality of the Mayer-Vietoris sequences for the constrained cohomology with respect to morphisms of hypergraphs induced by bijective maps between the vertices. Consequently, we obtain the Mayer-Vietoris sequences for the constrained persistent (co)homology for filtrations of hypergraphs. As applications, we propose the constrained persistent (co)homology as a tool for the computation of higher-dimensional persistent homology of large networks.