Topological Graph Persistence

TL;DR

This paper explores advanced topological methods to extract hidden information from graphs, extending homological persistence techniques with novel graph-theoretical constructions like independent sets and neighborhoods.

Contribution

It introduces new topological constructions for graphs, including independent sets and enclaveless sets, to enhance the analysis of their persistent homology.

Findings

New constructions reveal hidden topological features in graphs.

Extended persistence methods provide deeper insights into graph structure.

Approach generalizes previous homological persistence techniques.

Abstract

Graphs are a basic tool for the representation of modern data. The richness of the topological information contained in a graph goes far beyond its mere interpretation as a one-dimensional simplicial complex. We show how topological constructions can be used to gain information otherwise concealed by the low-dimensional nature of graphs. We do that by extending previous work of other researchers in homological persistence, by proposing novel graph-theoretical constructions. Beyond cliques, we use independent sets, neighborhoods, enclaveless sets and a Ramsey-inspired extended persistence.