Local Global Geometric Insights for Graph Neural Networks via Entropic Curvature

Curvature notions on graphs, particularly Ollivier–Ricci and Forman curvatures, have emerged as powerful tools for addressing fundamental issues in Graph Neural Networks (GNNs), such as oversmoothing and oversquashing. While these metrics capture the stability and efficiency of information propagation, they primarily rely on local neighborhood comparisons and may overlook broader global features. In contrast, we introduce Entropic Curvature, which extends the Lott–Sturm–Villani framework to discrete spaces by focusing on the global displacement of probability measures along geodesics in Wasserstein space. By relating entropy, optimal transport, and curvature lower bounds, this framework offers powerful functional-analytic tools, such as transport-entropy and Poincaré-type inequalities, which establish a thermodynamic regularity on the information flow. We prove that positive entropic curvature provides quantitative guarantees against feature collapse and adversarial distribution shifts. Furthermore, we establish a fundamental expansion paradox, proving that sparsity, strong spectral expansion, and positive entropic curvature cannot coexist in large graphs, revealing that the fast communication required to prevent oversquashing inherently forces a move toward negative curvature and instability. Empirically, we introduce a tractable Weak Entropic Curvature proxy and an E-Gate mechanism that modulates message-passing based on this global prior, consistently enhancing training stability and performance across diverse benchmarks.