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Accelerated Evaluation of Ollivier-Ricci Curvature Lower Bounds: Bridging Theory and Computation

Wonwoo Kang, Heehyun Park

TL;DR

This work addresses the computational bottleneck of estimating Ollivier-Ricci curvature (ORC) by deriving a generalized, laziness-aware lower bound that applies to spaces with integer-valued metrics and discrete random walks. It then delivers a linear-time algorithm to bound the Wasserstein distance $W_1$ between local measures, enabling scalable ORC computation, and extends the framework to hypergraphs via aggregation functions (AGG$_A$, AGG$_B$, AGG$_M$). Through theoretical development and empirical validation on synthetic and real-world datasets, the method achieves substantial speedups over traditional Sinkhorn-based approaches while preserving the informative structure of curvature for tasks like community detection. The approach broadens applicability to large networks and complex hypergraph structures, providing a practical tool for curvature-based network analysis.

Abstract

Curvature serves as a potent and descriptive invariant, with its efficacy validated both theoretically and practically within graph theory. We employ a definition of generalized Ricci curvature proposed by Ollivier, which Lin and Yau later adapted to graph theory, known as Ollivier-Ricci curvature (ORC). ORC measures curvature using the Wasserstein distance, thereby integrating geometric concepts with probability theory and optimal transport. Jost and Liu previously discussed the lower bound of ORC by showing the upper bound of the Wasserstein distance. We extend the applicability of these bounds to discrete spaces with metrics on integers, specifically hypergraphs. Compared to prior work on ORC in hypergraphs by Coupette, Dalleiger, and Rieck, which faced computational challenges, our method introduces a simplified approach with linear computational complexity, making it particularly suitable for analyzing large-scale networks. Through extensive simulations and application to synthetic and real-world datasets, we demonstrate the significant improvements our method offers in evaluating ORC.

Accelerated Evaluation of Ollivier-Ricci Curvature Lower Bounds: Bridging Theory and Computation

TL;DR

This work addresses the computational bottleneck of estimating Ollivier-Ricci curvature (ORC) by deriving a generalized, laziness-aware lower bound that applies to spaces with integer-valued metrics and discrete random walks. It then delivers a linear-time algorithm to bound the Wasserstein distance between local measures, enabling scalable ORC computation, and extends the framework to hypergraphs via aggregation functions (AGG, AGG, AGG). Through theoretical development and empirical validation on synthetic and real-world datasets, the method achieves substantial speedups over traditional Sinkhorn-based approaches while preserving the informative structure of curvature for tasks like community detection. The approach broadens applicability to large networks and complex hypergraph structures, providing a practical tool for curvature-based network analysis.

Abstract

Curvature serves as a potent and descriptive invariant, with its efficacy validated both theoretically and practically within graph theory. We employ a definition of generalized Ricci curvature proposed by Ollivier, which Lin and Yau later adapted to graph theory, known as Ollivier-Ricci curvature (ORC). ORC measures curvature using the Wasserstein distance, thereby integrating geometric concepts with probability theory and optimal transport. Jost and Liu previously discussed the lower bound of ORC by showing the upper bound of the Wasserstein distance. We extend the applicability of these bounds to discrete spaces with metrics on integers, specifically hypergraphs. Compared to prior work on ORC in hypergraphs by Coupette, Dalleiger, and Rieck, which faced computational challenges, our method introduces a simplified approach with linear computational complexity, making it particularly suitable for analyzing large-scale networks. Through extensive simulations and application to synthetic and real-world datasets, we demonstrate the significant improvements our method offers in evaluating ORC.
Paper Structure (28 sections, 13 theorems, 59 equations, 2 figures, 1 algorithm)

This paper contains 28 sections, 13 theorems, 59 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Backgrounds
  3. Graph Theory
  4. Metric Spaces on Integers
  5. Ollivier's Ricci Curvature
  6. Lower Bound of ORC
  7. Lower Bound of ORC on Spaces with Metrics on Integers
  8. Curvature with Laziness
  9. Lower bounds of ORC
  10. Algorithm
  11. Application: Curvature in Hypergraphs
  12. Experiments
  13. Time Evaluation
  14. Curvature Evaluation
  15. Limitations and Conclusions
  16. ...and 13 more sections

Key Result

Proposition 1

where the supremum is taken over all 1-Lipschitz continuous functions on $G = (V, E)$, i.e., functions $f: V \rightarrow \mathbb{R}$ such that for any $x, y \in V$, $x \neq y$.

Figures (2)

  • Figure 1: Logarithmic scale graph showing the time evaluation for computing the Wasserstein distance across different datasets.
  • Figure 2: Evaluations of curvatures

Theorems & Definitions (21)

  • Definition 1: Ollivier's Ricci curvature
  • Proposition 1: Kantorovich Duality
  • Theorem 1
  • Theorem 2
  • Corollary 1
  • Theorem 3
  • Theorem 3
  • Remark
  • proof
  • Theorem 3
  • ...and 11 more