Cluster-Persistence for Weighted Graphs

TL;DR

This paper introduces a novel persistent homology-based filtration for weighted graphs that enhances cluster analysis by providing richer signatures, robustness to outliers, and computational efficiency.

Contribution

It presents a new filtration method for persistent homology that improves cluster analysis in weighted graphs by incorporating non-trivial birth times and outlier robustness.

Findings

Provides richer signatures for connected components

Demonstrates computational efficiency and practical effectiveness

Explores properties on random graphs

Abstract

Persistent homology is a natural tool for probing the topological characteristics of weighted graphs, essentially focusing on their $0$-dimensional homology. While this area has been substantially studied, we present a new approach to constructing a filtration for cluster analysis via persistent homology. The key advantages of the new filtration is that (a) it provides richer signatures for connected components by introducing non-trivial birth times, and (b) it is robust to outliers. The key idea is that nodes are ignored until they belong to sufficiently large clusters. We demonstrate the computational efficiency of our filtration, its practical effectiveness, and explore into its properties when applied to random graphs.