Entropic Curvature is introduced, a global, transport-based curvature obtained by extending the Lott-Sturm-Villani framework to graphs through the displacement convexity of entropy along Wasserstein geodesics, and an expansion paradox proving that sparsity, strong spectral expansion, and positive entropic curvature cannot coexist in large graphs is proved.
Abstract
Curvature notions on graphs, particularly Ollivier-Ricci and Forman, have emerged as powerful tools for addressing fundamental issues in Graph Neural Networks (GNNs) such as oversmoothing and oversquashing, but rely almost exclusively on local edge-level comparisons and therefore fail to certify how information actually propagates over long distances. We introduce Entropic Curvature, a global, transport-based curvature obtained by extending the Lott-Sturm-Villani framework to graphs through the displacement convexity of entropy along Wasserstein geodesics. We define a tractable Weak Entropic Curvature proxy that lower-bounds the global entropic curvature, and from it derive (i) a Poincare-type inequality controlling oversmoothing, (ii) a transport-entropy generalization bound, and (iii) an expansion paradox proving that sparsity, strong spectral expansion, and positive entropic curvature cannot coexist in large graphs, unifying oversmoothing and oversquashing as opposite ends of a single curvature spectrum. We translate the theory into three practical mechanisms, the E-Gate aggregator, the ENT structural encoding, and Midpoint-Completion Rewiring (MCR), and benchmark them against SDRF, FoSR, BORF, LCP, and Graph Ricci Flow on six node-classification benchmarks, and graph-classification.
This work introduces a principled extension of Ollivier's Ricci curvature to complex-weighted graphs, which encompasses directed graphs as a special case and establishes fundamental theoretical properties of this new notion, including relations to the magnetic Laplacian and combinatorial upper and lower bounds that relate curvature to cycle structure in local neighborhoods.
Yu Tian, Eleanor P. Wiesler, Melanie Weber· 0 citations
ESNN is introduced, an Equivariant Sheaf Neural Network that enriches this interaction by learning directed, matrix-valued transport between neighboring vector features while preserving exact Euclidean equivariance.
Alessio Borgi, M. Severino, Fabrizio Silvestri et al.· 0 citations
Non-Euclidean spaces inherently enable high-fidelity embeddings for hierarchical and cyclical data due to their geometric properties. Existing approaches unify hyperbolic and spherical embeddings within the framework of constant curvature spaces. However, current methods for Lipschitz regularization remain limited to non-positive curvature geometries, such as hyperbolic and Euclidean spaces, and cannot be naturally extended to the general constant curvature setting. In this paper, we present a rigorous Lipschitz analysis for constant curvature graph convolutional networks ( \(\kappa\) -GCNs) and enhance their robustness through Lipschitz regularization. We derive upper bounds for the Lipschitz constants across constant curvature spaces, thereby standardizing the Lipschitz limits of the \(\kappa\) -stereographic model. Furthermore, we incorporate these bounds into a regularization framework for \(\kappa\) -GCNs to improve stability and robustness. Experimental results demonstrate that the proposed regularization method often strengthens the robustness of \(\kappa\) -GCNs across various curvature regimes, particularly under Gaussian feature noise.
Yang Shi, Jingchao Wang, Liangsi Lu et al.· ACM Transactions on Knowledg...· 0 citations
The findings illustrate that geometric insights grounded in hyperbolic geometry can offer powerful tools for understanding, embedding, and visualizing complex graph structures.
†. SalouaNaama, Kave Salamatian, M. Crovella· 0 citations
Taking classical information geometry as its point of departure, this paper investigates, through gradient flows, how dually flat geometry extends beyond regular convexity, non-degeneracy, and smoothness. The regular theory is developed from the log-determinant potential on positive definite Gram matrices, establishing its Legendre dual, Fisher--Rao metric, Bregman divergence, and generalized Pythagorean theorem. We connect this framework to Craig--Sakamoto deformation, Wolfe duality, and, via Yoshizawa's embedding, Brockett--Bloch--Ratiu double-bracket flows, linking isospectral dynamics, Stiefel optimization, and component learning. The Bures--Wasserstein geometry provides a complementary gradient-flow structure. The singular theory emerges from boundary behavior: difference-of-convex deformations produce indefinite or degenerate Hessians while retaining pseudo-Hessian, dually flat, Legendre-self-dual structures. Newton flows exhibit finite-time collapse or {\L}ojasiewicz-controlled convergence near non-Morse critical sets. Fisher-metric degeneracies on the Birkhoff polytope and elliptic-curve moduli are resolved by explicit blow-ups, yielding a birationally invariant exponential decay law. We further derive a closed-form Kirillov Jacobian and introduce cross curvature as a spectral diagnostic of local escape rates, including a new Box--Cox interpolation. Reproducible numerical experiments support the closed-form results. Rather than claiming a completed theory, the paper provides foundations for singular information geometry centered on degenerate pencils, indefinite dual flatness, blow-up geometry, and {\L}ojasiewicz-type convergence.
The Gromov-Wasserstein (GW) distance provides a principled framework for aligning metric measure (mm) spaces based solely on their intrinsic structure. Its ability to identify isomorphic representations of distributions across spaces renders it valuable for comparing data where equality up to isomorphism occurs naturally such as in graphs or, more generally, distributions on graphs. Recently, a type of dual form for the GW distance between Euclidean distributions with the squared Euclidean or inner product costs was derived, spurring the development of new statistical and algorithmic results for this setting. This work furnishes a novel duality result for GW distances with and without entropic regularization that is applicable to all finitely supported mm spaces. Leveraging this result, we derive the sample complexity of empirical GW distances between finite mm spaces, as well as limit distributions under proper centering and scaling. Furthermore, we propose new algorithms for solving the regularized GW problem which are subject to formal convergence guarantees. These statistical and algorithmic advancements give rise to a principled and efficient framework for testing whether two distributions on the set of graphs with a fixed number of nodes are isomorphic based on samples.
Gabriel Rioux, Joanna Marks, Riccardo Passeggeri et al.· 0 citations
We use cookies to run the site and, with your consent, for analytics and to show ads.
See our Cookie Policy.
