Weighted Persistent Homology

TL;DR

This paper extends persistent homology to weighted simplicial complexes with weights in an integral domain, enabling the distinction of filtrations that ordinary persistent homology cannot differentiate, especially when specific points are considered special.

Contribution

It generalizes the theory of weighted persistent homology to integral domains and demonstrates its ability to distinguish filtrations that ordinary persistent homology cannot.

Findings

Weighted persistent homology can distinguish filtrations with special points.

The theory is generalized to weights in an integral domain.

Weighted persistent homology captures additional topological information.

Abstract

In this paper we develop the theory of weighted persistent homology. In 1990, Robert J. Dawson was the first to study in depth the homology of weighted simplicial complexes. We generalize the definitions of weighted simplicial complex and the homology of weighted simplicial complex to allow weights in an integral domain $R$. Then we study the resulting weighted persistent homology. We show that weighted persistent homology can tell apart filtrations that ordinary persistent homology does not distinguish. For example, if there is a point considered as special, weighted persistent homology can tell when a cycle containing the point is formed or has disappeared.