Sobolev-Ricci 曲率
Sobolev--Ricci Curvature
AI总结：
本文提出Sobolev-Ricci曲率，通过Sobolev传输几何诱导出图曲率，具备高效评估能力，适用于图结构的曲率驱动变换和流形导向的边修剪。
摘要：
Ricci曲率是微分几何中编码局部几何结构的基本概念，其图论类比近期作为重加权、修剪和重塑网络几何的实用工具受到重视。我们提出Sobolev-Ricci曲率（SRC），通过Sobolev传输几何自然诱导出图Ricci曲率，可通过邻域测度上的树度Sobolev结构高效评估。我们建立了两项一致性行为，将SRC与经典传输曲率联系起来：(i) 在带有长度测度的树上，SRC在W1设置下恢复Ollivier-Ricci曲率（ORC）；(ii) SRC在Dirac极限下消失，匹配测度理论Ricci曲率的平坦情况。我们展示了SRC作为可重用曲率原始构件在两个代表性流程中的应用。我们定义Sobolev-Ricci流，通过在Ricci-流风格的重加权规则中用SRC替代ORC，并使用SRC进行曲率引导的边修剪以保持流形结构。总体而言，SRC为可扩展的曲率驱动图变换和流形导向的修剪提供了传输基础。
英文摘要：
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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