A Residual-Shell-Based Lower Bound for Ollivier-Ricci Curvature

2026-04-13

AI Summary

This work addresses the practical limitations of Ollivier–Ricci curvature, which relies on the computationally expensive Wasserstein distance. Existing low-complexity lower-bound approximations suffer from limited accuracy and are restricted to 1-hop random walks. To overcome these issues, the authors propose a novel lower bound based on residual shells that, for the first time, supports k-hop random walks. This approach substantially reduces computational complexity while significantly improving approximation accuracy. The method is both general and tight, achieving speedups of up to an order of magnitude on various fundamental graph structures and markedly narrowing the gap with exact curvature values—thus effectively balancing efficiency and fidelity.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Machine Learning: Graph-based Machine Learning
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

Ollivier-Ricci curvature (ORC), defined via the Wasserstein distance that captures rich geometric information, has received growing attention in both theory and applications. However, the high computational cost of Wasserstein distance evaluation has significantly limited the broader practical use of ORC. To alleviate this issue, previous work introduced a computationally efficient lower bound as a proxy for ORC based on 1-hop random walks, but this approach empirically exhibits large gaps from the exact ORC. In this paper, we establish a substantially tighter lower bound for ORC than the existing lower bound, while retaining much lower computational cost than exact ORC computation, with practical speedups of tens of times. Moreover, our bound is not restricted to 1-hop random walks, but also applies to k-hop random walks (k > 1). Experiments on several fundamental graph structures demonstrate the effectiveness of our bound in terms of both approximation accuracy and computational efficiency.

Problem

• Ollivier-Ricci curvature
• Wasserstein distance
• computational cost
• lower bound
• random walks

Innovation

• Ollivier-Ricci curvature
• Wasserstein distance
• lower bound
• random walks
• graph geometry