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

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 (Formula presented.), 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.