Spectral and Homological Bounds on k-Component Edge Connectivity

TL;DR

This paper introduces a new theoretical framework linking spectral graph theory and homology to analyze k-component edge connectivity, providing bounds on network resilience in complex systems.

Contribution

It extends classical edge connectivity to simplicial complexes and derives spectral-homological bounds, offering a multi-dimensional approach to network robustness analysis.

Findings

Derived tight spectral-homological bounds on edge removal

Applied framework to real-world infrastructure networks

Extended analysis to random and expander graphs

Abstract

We present a novel theoretical framework connecting k-component edge connectivity with spectral graph theory and homology theory to pro vide new insights into the resilience of real-world networks. By extending classical edge connectivity to higher-dimensional simplicial complexes, we derive tight spectral-homological bounds on the minimum number of edges that must be removed to ensure that all remaining components in the graph have size less than k. These bounds relate the spectra of graph and simplicial Laplacians to topological invariants from homology, establishing a multi-dimensional measure of network robustness. Our framework improves the understanding of network resilience in critical systems such as the Western U.S. power grid and European rail network, and we extend our analysis to random graphs and expander graphs to demonstrate the broad applicability of the method. Keywords: k-component edge connectivity, spectral graph theory, homology, simplicial complexes, network resilience, Betti numbers, algebraic connectivity, random graphs, expander graphs, infrastructure systems