From Persistence to Resilience: New Betti Numbers for Analyzing Robustness in Simplicial Complex Networks

TL;DR

This paper introduces new Betti numbers inspired by persistent homology to quantify the robustness and structural properties of simplicial complex networks, enabling better analysis of their resilience to failures.

Contribution

It proposes thick and cohesive Betti numbers as novel invariants that capture higher-order structural information and resilience in simplicial complexes, extending classical topological descriptors.

Findings

New Betti numbers effectively measure cycle thickness and connection strength.

Framework for analyzing robustness via biparameter persistence modules.

Deeper understanding of structural dynamics in higher-order networks.

Abstract

Persistent homology is a fundamental tool in topological data analysis; however, it lacks methods to quantify the fragility or fineness of cycles, anticipate their formation or disappearance, or evaluate their stability beyond persistence. Furthermore, classical Betti numbers fail to capture key structural properties such as simplicial dimensions and higher-order adjacencies. In this work, we investigate the robustness of simplicial networks by analyzing cycle thickness and their resilience to failures or attacks. To address these limitations, we draw inspiration from persistent homology to introduce filtrations that model distinct simplicial elimination rules, leading to the definition of two novel Betti number families: thick and cohesive Betti numbers. These improved invariants capture richer structural information, enabling the measurement of the thickness of the links in the homology cycle and the assessment of the strength of their connections. This enhances and refines classical topological descriptors and our approach provides deeper insights into the structural dynamics of simplicial complexes and establishes a theoretical framework for assessing robustness in higher-order networks. Finally, we establish that the resilience of topological features to simplicial attacks can be systematically examined through biparameter persistence modules, wherein one parameter encodes the progression of the attack, and the other captures structural refinements informed by thickness or cohesiveness.