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

TL;DR

This paper introduces a tighter, computationally efficient lower bound for Ollivier-Ricci curvature that improves approximation accuracy and extends to k-hop random walks, enabling faster analysis of graph structures.

Contribution

The authors develop a new residual-shell-based lower bound for ORC that is more accurate than previous bounds and applicable to multiple hop lengths, with significant speed improvements.

Findings

The new bound is substantially tighter than existing lower bounds.

It achieves practical speedups of tens of times over exact ORC computation.

Experiments show improved approximation accuracy and efficiency on various graph structures.

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.