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Discrete scalar curvature as a weighted sum of Ollivier-Ricci curvatures

Abigail Hickok, Andrew J. Blumberg

TL;DR

This paper defines scalar Ollivier-Ricci curvature (SORC) as a weighted node-wise sum of edge Ollivier-Ricci curvatures on graphs, and proves that, under sampling from a compact Riemannian manifold and a proper scaling, SORC converges in probability to the manifold's scalar curvature $S(x)$. The authors decompose the convergence into bounds relating graph and manifold Wasserstein distances and to Ricci curvature, establishing two new results on ORC convergence to Ricci: a non-asymptotic edge-wise bound and a probabilistic edge convergence. A key methodological contribution is adapting Hutchinson's trace trick to estimate the Ricci trace via a local log-map construction, enabling node-wise convergence results. Theoretical findings are complemented by numerical experiments on random geometric graphs drawn from spheres, demonstrating the proposed scaling and convergence behavior, and the paper discusses practical pre-processing steps for applying the approach to real-world networks.

Abstract

We study the relationship between discrete analogues of Ricci and scalar curvature that are defined for point clouds and graphs. In the discrete setting, Ricci curvature is replaced by Ollivier-Ricci curvature. Scalar curvature can be computed as the trace of Ricci curvature for a Riemannian manifold; this motivates a new definition of a scalar version of Ollivier-Ricci curvature. We show that our definition converges to scalar curvature for nearest neighbor graphs obtained by sampling from a manifold. We also prove some new results about the convergence of Ollivier-Ricci curvature to Ricci curvature.

Discrete scalar curvature as a weighted sum of Ollivier-Ricci curvatures

TL;DR

This paper defines scalar Ollivier-Ricci curvature (SORC) as a weighted node-wise sum of edge Ollivier-Ricci curvatures on graphs, and proves that, under sampling from a compact Riemannian manifold and a proper scaling, SORC converges in probability to the manifold's scalar curvature . The authors decompose the convergence into bounds relating graph and manifold Wasserstein distances and to Ricci curvature, establishing two new results on ORC convergence to Ricci: a non-asymptotic edge-wise bound and a probabilistic edge convergence. A key methodological contribution is adapting Hutchinson's trace trick to estimate the Ricci trace via a local log-map construction, enabling node-wise convergence results. Theoretical findings are complemented by numerical experiments on random geometric graphs drawn from spheres, demonstrating the proposed scaling and convergence behavior, and the paper discusses practical pre-processing steps for applying the approach to real-world networks.

Abstract

We study the relationship between discrete analogues of Ricci and scalar curvature that are defined for point clouds and graphs. In the discrete setting, Ricci curvature is replaced by Ollivier-Ricci curvature. Scalar curvature can be computed as the trace of Ricci curvature for a Riemannian manifold; this motivates a new definition of a scalar version of Ollivier-Ricci curvature. We show that our definition converges to scalar curvature for nearest neighbor graphs obtained by sampling from a manifold. We also prove some new results about the convergence of Ollivier-Ricci curvature to Ricci curvature.

Paper Structure

This paper contains 16 sections, 36 theorems, 179 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Background and Assumptions
  3. Riemannian geometry and curvature
  4. Random geometric graphs
  5. Ollivier-Ricci curvature
  6. The mean Ricci curvature at a node converges to scalar curvature
  7. Convergence rate of Ollivier-Ricci graph curvature
  8. Bounding $|W_1^G(\mu_x^G, \mu_x^G) - W_1^M(\mu_x^G, \mu_y^G)|$
  9. Bounding $|W_1^M(\mu_x^G, \mu_x^G) - W_1^M(\mu_x^M, \mu_y^M)|$
  10. Bounds on the error between Ollivier-Ricci curvature and Ricci curvature
  11. Main Results
  12. The ratio $\epsilon / d_M(x, y)$
  13. Proof that scalar Ollivier-Ricci curvature converges to scalar curvature
  14. Numerical Experiments
  15. Conclusions
  16. ...and 1 more sections

Key Result

Theorem 1.1

As the number of nodes $N \to \infty$ and the connection threshold $\epsilon_N \to 0$ (at an appropriate rate) in a random geometric graph $G_N$, we have for a random node $x_N \in G_N$.

Figures (2)

  • Figure 1: Edges (in red) with (A) negative ORC, (B) zero ORC, and (C) positive ORC. We set each edge weight to $1$ and calculate $\kappa_G(x, y)$ using the formulation in \ref{['def:orc']}.
  • Figure 2: The graphs are RGGs with nodes sampled from spheres of dimensions $n \in \{2, 3, 4\}$. The solid lines are the mean $\frac{2(n+2)^2}{\epsilon_N^4}\text{SORC}(x)$ as a function of the number $N$ of nodes in the graph. The dashed lines are the scalar curvatures of the corresponding spheres. The connectivity threshold $\epsilon_N$ is set to $C_n N^{-\alpha_n}$, where $\alpha_n = \frac{1}{6.01n}$ so that $\alpha$ satisfies our condition $\alpha \in (0, \frac{1}{6n})$. In (A), the constant $C_n$ is set so that the average degree at $N = 1000$ is approximately $50$. In (B), the constant $C_n$ is set so that the average degree at $N = 1000$ is approximately $100$. More precisely, if $k$ is the desired average degree at $N = 1000$, we set $C_n$ such that $k = N \frac{v_n \epsilon_N^n}{\text{vol}(S^n)}$, which is approximately equal to the expected degree of a node, ignoring curvature.

Theorems & Definitions (66)

  • Theorem 1.1: \ref{['thm:main']}
  • Proposition 1.2: \ref{['prop:orc_convergence1']}
  • Proposition 1.3
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • Definition 2.4: Ollivier-Ricci curvature on a graph
  • Remark 2.5
  • Theorem 2.6: ORC_convergence_hoorn
  • Theorem 2.7: ORC_convergence_hoorn
  • ...and 56 more