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

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

Mayer-Vietoris Sequences for the Constrained Persistent Homology of Simplicial Complexes

The notion of independence hypergraphs is introduced to investigate the relations between random hypergraphs and random simplicial complexes [29]. With the help of the differential calculus on discrete sets, the constrained homology of simplicial complexes as well as the constrained cohomology of independence hypergraphs are constructed [26]. In this paper, by proving the functorialities of the constrained homology of simplicial complexes as well as the constrained cohomology of independence hypergraphs, we study the constrained persistent homology for filtrations of simplicial complexes as well as the constrained persistent cohomology for filtrations of independence hypergraphs. We study the Mayer-Vietoris sequences for the constrained (co)homology as well as their functorialities. As a result, we prove the Mayer-Vietoris sequences for the persistent constrained homology for filtrations of simplicial complexes and the persistent constrained cohomology for filtrations of independence hypergraphs

Operators on random hypergraphs and random simplicial complexes

Random hypergraphs and random simplicial complexes have potential applications in computer science and engineering. Various models of random hypergraphs and random simplicial complexes on n-points have been studied. Let L be a simplicial complex. In this paper, we study random sub-hypergraphs and random sub-complexes of L. By considering the minimal complex that a sub-hypergraph can be embedded in and the maximal complex that can be embedded in a sub-hypergraph, we define some operators on the space of probability functions on sub-hypergraphs of L. We study the compositions of these operators as well as their actions on the space of probability functions. As applications in computer science, we give algorithms generating large sparse random hypergraphs and large sparse random simplicial complexes.Comment: 22 page

Weighted (Co)homology and Weighted Laplacian

In this paper, we generalize the combinatorial Laplace operator of Horak and Jost by introducing the $\phi$-weighted coboundary operator induced by a weight function $\phi$. Our weight function $\phi$ is a generalization of Dawson's weighted boundary map. We show that our above-mentioned generalizations include new cases that are not covered by previous literature. Our definition of weighted Laplacian for weighted simplicial complexes is also applicable to weighted/unweighted graphs and digraphs.Comment: 22 page