Sobolev--Ricci Curvature

• URL: http://arxiv.org/abs/2603.12652v1
• Date: Fri, 13 Mar 2026 04:50:36 GMT
• Title: Sobolev--Ricci Curvature
• Abstract summary: Ricci curvature is a fundamental concept in differential geometry for encoding local geometric structure.<n>We propose Sobolev-Ricci Curvature (SRC), a graph Ricci curvature induced by Sobolev transport geometry.
• Score: 12.001517547230838
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Ricci curvature is a fundamental concept in differential geometry for encoding local geometric structure, and its graph-based analogues have recently gained prominence as practical tools for reweighting, pruning, and reshaping network geometry. We propose Sobolev-Ricci Curvature (SRC), a graph Ricci curvature canonically induced by Sobolev transport geometry, which admits efficient evaluation via a tree-metric Sobolev structure on neighborhood measures. We establish two consistency behaviors that anchor SRC to classical transport curvature: (i) on trees endowed with the length measure, SRC recovers Ollivier-Ricci curvature (ORC) in the canonical W1 setting, and (ii) SRC vanishes in the Dirac limit, matching the flat case of measure-theoretic Ricci curvature. We demonstrate SRC as a reusable curvature primitive in two representative pipelines. We define Sobolev-Ricci Flow by replacing ORC with SRC in a Ricci-flow-style reweighting rule, and we use SRC for curvature-guided edge pruning aimed at preserving manifold structure. Overall, SRC provides a transport-based foundation for scalable curvature-driven graph transformation and manifold-oriented pruning.

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