Topological Resonance in Stochastic Block Models: Phase Transition and Information Localization

Abstract

We study information propagation in structured networks using a continuous diffusion equation on graphs with weighted edges. The edge weight between two nodes is enhanced by a parameter β if they share a topological label (community membership) in a Stochastic Block Model (SBM). We find that for a critical value β_c = 1.2, the propagation time within the community reaches a minimum, while the inter-community propagation time is maximally delayed, yielding a resonance index ratio = 161.5 — meaning that information spreads 161 times faster inside the community than between communities. For β > β_c, a phase transition occurs, and information bursts through the inter-community barriers. This effect is interpreted as a topological resonance, analogous to the resonance phenomena observed in previous applications of the Roudy Protocol to clinical medicine, structural biology, and quantum physics. We propose that this model provides a rigorous mathematical foundation for the protocol and suggests applications in network dynamics, epidemiology, and social systems.