When Single-Scale Betti Counts Are Not Enough: Ring Statistics for Structured Network Populations

Abstract

How can one test for a multiplicative topological difference between two structured network populations whose fixed-scale additive Betti summaries agree? We model each population as a probability law over finite graphs, considered up to isomorphism, and read each graph through its clique complex. At the working scale, the ordinary summary is the joint Betti vector B=(b0,b1,b2), recording connected components, loops, and voids. The comparison is deliberately single scale: B is the vector of Betti counts at a fixed working scale, not the full persistence diagram of a filtration. We show that this additive summary can be identical under two non-degenerate graph laws while a multiplicative cohomology-ring statistic differs: the cup product, a multiplicative operation recording when two one-dimensional cohomology classes have a nonzero product in degree two, occurs with different frequency under the two laws. Consequently any procedure whose input is only this single-scale B-summary has no power beyond its size against the constructed alternatives, while a simple cup-product statistic separates them. We define a ring-frequency distance, prove finite-sample concentration for its plug-in estimator, and give a consistent two-sample test. The theory is aimed at structured, ring-rich graph complexes—surface-like meshes and coverage complexes—where the cup product is active; we give a deterministic mechanism under which it is vacuous, together with numerical evidence that it can be uninformative in generic random-graph regimes. Numerical illustrations confirm that the ring test detects the difference while calibrated B-only tests stay blind; those same B-only tests have power when the Betti law itself changes. Real-data case studies on surface meshes and nanoporous frameworks illustrate, respectively, the intended ring-rich regime and a cup-vacuous scope boundary.