Weighted Persistent Homology

TL;DR

This paper introduces weighted versions of classical topological complexes, demonstrating their stability and potential applications, thereby extending persistent homology tools to weighted data.

Contribution

It presents weighted cech and Vietoris-Rips complexes, establishing their stability and applicability, which is a novel extension of persistent homology methods.

Findings

Weighted complexes satisfy a version of the Vietoris-Rips Lemma.

Weighted complexes exhibit stability properties.

Preliminary applications demonstrate their usefulness.

Abstract

We introduce weighted versions of the classical \v{C}ech and Vietoris-Rips complexes. We show that a version of the Vietoris-Rips Lemma holds for these weighted complexes and that they enjoy appropriate stability properties. We also give some preliminary applications of these weighted complexes.