# Sobolev--Ricci Curvature

arXiv: https://arxiv.org/abs/2603.12652 · HTML: https://mlsanity.com/paper/2603.12652/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 17 of its 37 references that are in the map.
- First posted on arXiv 13 Mar 2026.

## Builds on (most cited first, 17 of 17)

- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Understanding over-squashing and bottlenecks on graphs via curvature](https://mlsanity.com/paper/2111.14522/) (2021)
- [Generalized Sliced Wasserstein Distances](https://mlsanity.com/paper/1902.00434/) (2019)
- [Community Detection on Networks with Ricci Flow](https://mlsanity.com/paper/1907.03993/) (2019)
- [Tree-Sliced Variants of Wasserstein Distances](https://mlsanity.com/paper/1902.00342/) (2019)
- [Manifold Regularized Dynamic Network Pruning](https://mlsanity.com/paper/2103.05861/) (2021)
- [Heterogeneous manifolds for curvature-aware graph embedding](https://mlsanity.com/paper/2202.01185/) (2022)
- [Ricci Curvature of the Internet Topology](https://mlsanity.com/paper/1501.04138/) (2015)
- [Sobolev Transport: A Scalable Metric for Probability Measures with Graph Metrics](https://mlsanity.com/paper/2202.10723/) (2022)
- [Curvature-based Clustering on Graphs](https://mlsanity.com/paper/2307.10155/) (2023)
- [Effective Structural Encodings via Local Curvature Profiles](https://mlsanity.com/paper/2311.14864/) (2023)
- [Ollivier-Ricci curvature convergence in random geometric graphs](https://mlsanity.com/paper/2008.01209/) (2020)
- [Manifold-aligned Neighbor Embedding](https://mlsanity.com/paper/2205.11257/) (2022)
- [Recovering Manifold Structure Using Ollivier-Ricci Curvature](https://mlsanity.com/paper/2410.01149/) (2024)
- [An efficient alternative to Ollivier-Ricci curvature based on the Jaccard metric](https://mlsanity.com/paper/1710.01724/) (2017)
- [Accelerated Evaluation of Ollivier-Ricci Curvature Lower Bounds: Bridging Theory and Computation](https://mlsanity.com/paper/2405.13302/) (2024)
- [On computing Discretized Ricci curvatures of graphs: local algorithms and (localized) fine-grained reductions](https://mlsanity.com/paper/2208.09535/) (2022)

## Abstract (arXiv)

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.