Stability and Extension of Steady and Ranging Persistence

TL;DR

This paper extends steady and ranging persistence from graphs to other objects using category theory, analyzing their stability and providing practical hypergraph applications.

Contribution

It introduces a categorical extension of steady and ranging persistence and characterizes features that induce balanced persistence, with a practical hypergraph implementation.

Findings

Extended persistence concepts to broader objects via category theory.

Provided stability analysis for the extended persistence.

Demonstrated practical application on hypergraphs.

Abstract

Persistent homology is a topological data analysis tool that has been widely generalized, extending its scope beyond the field of topology. Among its extensions, steady and ranging persistence were developed to study a wide variety of graph properties. Precisely, given a feature of interest on graphs, it is possible to build two types of persistence (steady and ranging persistence) that follow the evolution of the feature along graph filtrations. This study extends steady and ranging persistence to other objects using category theory and investigates the stability of such persistence. In particular, a characterization of the features that induce balanced steady and ranging persistence is provided. The main results of this study are illustrated using a practical implementation for hypergraphs.