Local-Global Geometric Insights for Graph Neural Networks via Entropic Curvature

In “Local–Global Geometric Insights for Graph Neural Networks via Entropic Curvature,” Rachid Caich and Yassine Abbahaddou, from the Centre de recherches mathématiques at the University of Montréal and LIX at École Polytechnique, replace edge-local Ollivier–Ricci and Forman curvature with a global transport-based notion grounded in Lott–Sturm–Villani entropy convexity along W1-Wasserstein geodesics. Their tractable two-hop proxy, weak entropic curvature κw, provably lower-bounds global curvature and yields Poincaré-type control of oversmoothing, transport–entropy generalization bounds, and an expansion paradox: large sparse graphs cannot simultaneously have strong spectral expansion and positive curvature, unifying oversquashing and oversmoothing as opposite geometric regimes. The theory produces three practical tools: E-Gate, which modulates message aggregation by node curvature; ENT, a three-dimensional curvature encoding; and Midpoint-Completion Rewiring, or MCR, which adds alternative two-hop routes to bottlenecks. E-Gate improves the backbone in 16 of 20 tested backbone–dataset pairs at p < 0.05, while ENT leads competing encodings on 4 of 5 benchmarks. On Wisconsin, SAGE with E-Gate reaches 79.22 percent accuracy versus 75.10 percent for baseline SAGE. Across six random-graph families, MCR consistently beats Ollivier- and Forman-based rewiring, reaching a spectral-gap improvement of approximately 0.6 on stochastic block models after 10 added edges. Experiments use PyTorch Geometric and NVIDIA GPUs, including the RTX 3090.