Sobolev--Ricci Curvature

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.

Figures (13)

• Figure 1: Community detection performance (ARI) and computational cost under Ricci flow.
• Figure 2: The bars show the percentage of shortcut edges correctly removed (red) and the percentage of good edges incorrectly removed (light blue). Representative results on two datasets.
• Figure 3: Empirical root sensitivity of SRC(SPT) on SBM and LFR. For each dataset, we estimate $\mathbb{E}_{r,r'}[\|\Delta\kappa\|_1/|\Delta|]$ by sampling multiple root pairs $(r,r')$ and averaging over repeated graph instances, where $\|\Delta\kappa\|_1$ is the $L^1$ difference of edge curvatures and $|\Delta|=|E(T_r)\triangle E(T_{r'})|$ is the number of tree edges that change. This quantity corresponds to the left-hand side of \ref{['eq:expected_ratio_bound']} and measures the average curvature perturbation per changed tree edge.
• Figure 4: SRC(SPT) ablation on LFR: effect of the SRC exponent $p$ and downstream clustering. We evaluate $p\in\{1,1.5,2\}$ using two downstream procedures: Louvain on similarity weights (left) and threshold-based length-cut clustering (right). Top panels show standard deviation of ARI across trials, and bottom panels show the mean ARI. Louvain yields stable performance with little dependence on $p$, whereas length-cut is unstable and substantially degrades ARI.
• Figure 5: Ablation on the lazy random walk parameter $\alpha$ on LFR ($500$ nodes). We sweep $\alpha$ and plot the standard deviation (top row) and the mean (bottom row) of ARI over multiple random seeds as a function of the mixing parameter $\mu$. Left: SRC(SPT). Right: SRC(MST).

Theorems & Definitions (15)

• Definition 5.1: Length measure
• Proposition 5.2
• proof
• Theorem A.1: Root-dependence bound for SRC, $p=1$
• proof
• Lemma C.1: Stability of $\gamma_e$ under weak convergence
• proof
• Lemma C.2: Vanishing of $S_p$ to a Dirac
• proof
• Lemma C.3: Two basic facts for $W_1$