Abstract
The design of critical infrastructure faces a fundamental paradox: optimizing for efficiency systematically erodes resilience against cascading failures. While classical network theory attributes this to a lack of redundancy, we propose that resilience is fundamentally a topological property governed by Persistent Homology. We introduce the Topological Resilience Conjecture (TRC), asserting that for complex networks subject to monotone load-redistribution cascades, resilience R is predicted by a capacity-weighted persistent homology metric H_p, normalized by global efficiency E: R∝H_p/E. Unlike static Betti numbers, H_p integrates the lifetimes of H_1 cycles across a capacity-weighted filtration, capturing multi-scale structural integrity. We explicitly define the “monotone load-redistribution” universality class and address the “proxy problem” by demonstrating that H_p/E retains predictive power even when controlling for effective resistance, algebraic connectivity, and cycle rank. Using a rigorous pipeline based on Directed Flag Complexes, we simulate cascades across three network ensembles (N=500) matched for degree distribution, total capacity, and efficiency. While Ensembles A (Tree-dominant) and B (Random redundancy) fail under targeted attacks, Ensemble C (Optimized for multi-scale persistence) reduces cascade size by 68%. Multivariate regression confirms that H_p/E explains 34% of the variance in resilience independent of standard spectral and flow metrics (p<10^(−4)). This establishes persistence not merely as a proxy for redundancy, but as a distinct topological invariant governing shockwave containment. Historical and geopolitical analogies further illustrate the domain invariance of this universal framework.
