Computational Tools in Weighted Persistent Homology

TL;DR

This paper advances the theory and applications of weighted persistent homology by introducing new sequences and algorithms, enabling more efficient computation and analysis of weighted topological features.

Contribution

It introduces the Mayer-Vietoris sequence and generalized Bockstein spectral sequence for weighted homology, along with an algorithm for constructing filtrations from weighted networks.

Findings

Developed a new algorithm for weighted simplicial complex filtrations

Proved a theorem for calculating mod p^2 weighted persistent homology from mod p data

Extended the theoretical framework of weighted homology with new spectral sequences

Abstract

In this paper, we study further properties and applications of weighted homology and persistent homology. We introduce the Mayer-Vietoris sequence and generalized Bockstein spectral sequence for weighted homology. For applications, we show an algorithm to construct a filtration of weighted simplicial complexes from a weighted network. We also prove a theorem that allows us to calculate the mod $p^2$ weighted persistent homology given some information on the mod $p$ weighted persistent homology.