Sobolev--Ricci Curvature
This work provides a scalable curvature tool for graph analysis and transformation, addressing the need for efficient geometric methods in network applications, though it appears incremental by building on existing transport curvature concepts.
The authors introduced Sobolev-Ricci Curvature (SRC), a graph Ricci curvature based on Sobolev transport geometry, which enables efficient computation and is consistent with classical curvature measures. They demonstrated its utility in graph transformation tasks, such as Ricci-flow-style reweighting and curvature-guided edge pruning, to preserve manifold structure.
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.