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

Abstract

We show how persistence can be used to measure the robustness of cohomological cycles in finite simplicial complexes. We introduce two families of invariants: thick Betti numbers, which measure whether connected components and holes are supported by simplices of sufficiently high dimension; and cohesive Betti numbers, which measure the strength of higher-order adjacencies supporting cohomology classes. We then study robustness under simplicial degradation processes by combining an attack parameter with either the thick or the cohesive parameter, and by analyzing the resulting ladder modules through their image, kernel, and cokernel persistence modules. This allows us to distinguish features that remain structurally robust from those that lose thickness or cohesion during the degradation process. Finally, we prove stability results using the second network distance, which provides a theoretical reliability guarantee for the proposed constructions.