Sobolev--Ricci Curvature

TL;DR

The paper introduces Sobolev-Ricci Curvature (SRC), a new graph curvature measure based on Sobolev transport geometry, which aligns with classical curvature notions and enables scalable graph reweighting and pruning.

Contribution

It proposes SRC as a novel, efficient graph Ricci curvature derived from Sobolev transport geometry, connecting it to classical transport curvature and demonstrating its use in graph reweighting and pruning.

Findings

SRC recovers Ollivier-Ricci curvature on trees with length measure.

SRC vanishes in the Dirac limit, matching flat measure-theoretic Ricci curvature.

SRC enables curvature-guided edge pruning and manifold-preserving graph transformations.

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.