Beyond Pairwise Co-occurrence: Galois Closure Curvature for Higher-Order Relations

Hypergraphs represent interactions among arbitrary sets of entities, yet hypergraph learning methods can often consume incidence primarily through pairwise co-occurrence, i.e., which nodes appear together. This reduction can obscure set-level dependencies: two hypergraphs may share every node degree and pairwise co-occurrence count while differing in which node sets necessarily imply others. We ask how the geometry of such higher-order dependencies can be characterized beyond observed connectivity. Our key observation is that Galois closure, which encodes implications among nodes induced by incidence, reveals this geometry through how closure states contract or expand under shared structural perturbations, following the local contraction principle underlying Ollivier–Ricci curvature. Building on this observation, we introduce Galois Closure Curvature (GCC), which quantifies the perturbation response of closure states, together with a curvature profile that retains contraction, expansion, and variability otherwise lost through cancellation in the scalar curvature. We show that this profile can distinguish hypergraphs with identical pairwise co-occurrence statistics even when their scalar GCC coincides, and characterize conditions under which the closure response vanishes on purely pairwise structures. By lifting the profile to node–hyperedge incidences, GCC provides a context-dependent structural representation of higher-order implication geometry. Across controlled implication-recovery tasks and established hypergraph benchmarks, GCC achieves the best mean rank among curvature descriptors on Hy-WL and is the most accurate on collections whose classes differ only in twin collections. The code is available at https://anonymous.4open.science/r/GCC-7A1C/.