提出基于流形几何的图曲率新方法，可高效重连网络并保持数据结构。
Sobolev--Ricci Curvature
- 用舒尔夫传输几何构造图曲率，通过邻域度量树结构快速计算。
- 在树结构上等价于经典奥利维耶曲率，极限情况下曲率为零。
- 适用于网络重连与边剪枝，适合需要保持流形结构的研究者。
Ricci曲率是微分几何中刻画局部几何结构的核心概念，其图版本近年来成为重加权、剪枝和重塑网络几何的实用工具。本文提出舒尔夫-Ricci曲率（SRC），一种由舒尔夫传输几何自然诱导的图Ricci曲率，可通过邻域测度上的树度量舒尔夫结构实现高效计算。我们建立了两个一致性行为：(i) 在赋予长度测度的树上，SRC 在标准 W1 设置下恢复奥利维耶-Ricci曲率（ORC）；(ii) 在狄拉克极限下，SRC 为零，与测度论Ricci曲率的平坦情形一致。我们将SRC作为通用曲率基元应用于两个典型流程：定义了以SRC替代ORC的舒尔夫-Ricci流，用于类里奇流重连规则；并利用SRC进行曲率引导的边剪枝，以保留流形结构。总体而言，SRC为可扩展的曲率驱动图变换与面向流形的剪枝提供了传输几何基础。
原文摘要 · Abstract (English)
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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