Stable Homology-Based Cycle Centrality Measures

TL;DR

This paper introduces new cycle centrality measures based on algebraic topology and persistent homology, providing robust, multiscale insights into cycle importance in weighted graphs and complex point clouds.

Contribution

It develops novel homology-based cycle centrality measures utilizing persistent signatures, enhancing the analysis of higher-order structures in graphs and point clouds.

Findings

Measures are stable under small perturbations.

Effective in analyzing fractal-like point clouds.

Capture cycle importance beyond traditional topological summaries.

Abstract

Network centrality measures play a crucial role in understanding graph structures, assessing the importance of nodes, paths, or cycles based on directed or reciprocal interactions encoded by vertices and edges. Estrada and Ross extended these measures to simplicial complexes to account for higher-order connections. In this work, we introduce novel centrality measures by leveraging algebraically-computable topological signatures of cycles and their homological persistence. We apply tools from algebraic topology to extract multiscale signatures within cycle spaces of weighted graphs, tracking homology generators persisting across a weight-induced filtration of simplicial complexes built over point clouds. This approach incorporates persistent signatures and merge information of homology classes along the filtration, quantifying cycle importance not only by geometric and topological significance but also by homological influence on other cycles. We demonstrate the stability of these measures under small perturbations using an appropriate metric to ensure robustness in practical applications. Finally, we apply these measures to fractal-like point clouds, revealing their capability to detect information consistent with, and possibly overlooked by, common topological summaries.