cs.LGMar 13, 2026

Sobolev--Ricci Curvature

Authors: Kyoichi Iwasaki, Tam Le, Hideitsu Hino

Organizations: The Graduate University for Advanced Studies, SOKENDAI · The Institute of Statistical Mathematics

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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 & tables

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.
Appendix figures & tables14 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 3: Empirical root sensitivity of SRC(SPT) on SBM and LFR. For each dataset, we estimate Er,r′[∥Δκ∥1/∣Δ∣]\mathbb{E}_{r,r^{\prime}}[\|\Delta\kappa\|_{1}/|\Delta|] by sampling multiple root pairs (r,r′)(r,r^{\prime}) and averaging over repeated graph instances, where ∥Δκ∥1\|\Delta\kappa\|_{1} is the L1L^{1} difference of edge curvatures and ∣Δ∣=∣E(Tr)△E(Tr′)∣|\Delta|=|E(T_{r})\triangle E(T_{r^{\prime}})| is the number of tree edges that change. This quantity corresponds to the left-hand side of ( A.13 ) and measures the average curvature perturbation per changed tree edge.
Continuous geometry (Riemann/Finsler)Discrete Sobolev geometry (this paper)
Point x∈Mx\in MMeasure μ∈P(V)\mu\in\mathcal{P}(V)
Tangent space TxMT_{x}MTμP(V)={ξ∈RV  ;  ∑v∈Vξ(v)=0}(independent of μ).T_{\mu}\mathcal{P}(V)=\left\{\xi\in\mathbb{R}^{V}\;;\;\sum_{v\in V}\xi(v)=0\right\}\hskip 9.24994pt(\text{independent of }\mu).
Chart φ:U→Rd\varphi:U\to\mathbb{R}^{d}Cut-coordinate map Φz0:μ↦(μ(γe))e∈E\Phi_{z_{0}}:\mu\mapsto(\mu(\gamma_{e}))_{e\in E}
Riemannian metric ggp=2p=2 : weighted inner product ⟨a,b⟩λ=∑eλ(e)aebe\langle a,b\rangle_{\lambda}=\sum_{e}\lambda(e)a_{e}b_{e}
Finsler norm F(ξ)F(\xi)General pp : F(ξ)=∥dΦz0(ξ)∥ℓλpF(\xi)=\|d\Phi_{z_{0}}(\xi)\|_{\ell^{p}_{\lambda}}
Geodesic distanceSp(μ,ν)=∥Φz0(μ)−Φz0(ν)∥ℓλpS_{p}(\mu,\nu)=\|\Phi_{z_{0}}(\mu)-\Phi_{z_{0}}(\nu)\|_{\ell^{p}_{\lambda}}
Appendix
Table 1: Interpretative correspondence between continuous geometry and the discrete Sobolev geometry induced by an extracted tree.
ItemValue
Flow iterations TflowT_{\mathrm{flow}}2020 , early stop if Δκ(t)<ε\Delta\kappa^{(t)}<\varepsilon
Lazy RW parameter α\alpha0.50.5
ST exponent pp (SRC)11
Tree constructionMST / SPT
Root selection (SPT)00
Length-to-affinity β\beta1.01.0
Appendix
Table 2: Common hyperparameters used across community detection experiments for SRF.
Table 6
Figure 4: SRC(SPT) ablation on LFR: effect of the SRC exponent pp and downstream clustering. We evaluate p∈{1,1.5,2}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 pp , whereas length-cut is unstable and substantially degrades ARI.
Figure 5: Ablation on the lazy random walk parameter α\alpha on LFR ( 500500 nodes). We sweep α\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 μ\mu . Left: SRC(SPT). Right: SRC(MST).
Figure 6: Sensitivity to α\alpha on real-world networks. Error bars indicate std over runs.
Figure 7: LFR benchmark: sensitivity to the length-to-weight temperature β\beta in w~xy=exp⁡(−βwxy)\tilde{w}_{xy}=\exp(-\beta w_{xy}) . Left: ARI (mean ±\pm std) of SRC(SPT)+Louvain after the SRF. Right: best Louvain resolution selected by grid search (mean ±\pm std). Performance is stable for moderate β\beta (0.01–2), while overly large β\beta (5,10) collapses weights and degrades modularity-based clustering.
Figure 8: Ricci curvature distributions for different mixing parameters μ\mu .
Figure 9: Robustness to tree construction. SRC distributions on the same LFR graphs computed using different extracted trees (SPT, MST, Random spanning tree). Across μ\mu , the global distributional shapes are similar, suggesting SRC mainly reflects coarse graph geometry rather than a specific tree optimality criterion.
Figure 10: Two-dimensional synthetic datasets employed in the edge pruning experiments. Each dataset consists of 4,000 data points. Red lines indicate shortcut edges, i.e., edges that do not follow the underlying manifold structure.
Figure 11: Three-dimensional synthetic datasets employed in the edge pruning experiments. Each dataset contains 4,000 data points. Red lines indicate shortcut edges, i.e., edges that deviate from the underlying manifold structure.
Figure 12: Bars indicate the percentage of shortcut edges correctly removed (red) and good edges incorrectly removed (light blue). While Distance-only and ORC pruning often fail, “SRC only” achieves reliable shortcut detection with fewer mistakes. With MANL, ORC and SRC(SPT) consistently succeed.
Figure 13: Illustrative example on the 3D_swiss_roll dataset, where we see in Fig. 10 . Distribution of curvature values, showing that SRC(SPT) tends to assign larger curvature values than ORC. This leads to more effective identification of shortcut edges in the first-stage filtering.
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