Sobolev--Ricci Curvature

2026-03-13

AI Summary

This work proposes Sobolev–Ricci curvature (SRC), a novel notion of Ricci curvature on graphs that achieves both theoretical coherence and computational efficiency. Built upon Sobolev transport geometry, SRC leverages the tree-metric structure of neighborhood measures to efficiently approximate the Wasserstein-1 distance, enabling scalable curvature computation. The method exactly recovers Ollivier–Ricci curvature on trees and naturally degenerates to a flat limit under Dirac measures, thereby providing the first unified framework linking Sobolev transport geometry with graph curvature theory. Empirical results demonstrate that SRC serves as a versatile curvature primitive, effectively driving Sobolev–Ricci flow-based edge reweighting and curvature-guided edge pruning—enhancing geometric learning performance while preserving the underlying graph manifold structure.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Reasoning under Uncertainty: Graphical Models
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics

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.

Problem

• Ricci curvature
• graph geometry
• Sobolev transport
• manifold preservation
• network pruning

Innovation

• Sobolev-Ricci Curvature
• graph Ricci curvature
• optimal transport
• manifold-preserving pruning
• Ricci flow