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

TL;DR

This paper introduces a fast, linear-complexity method for evaluating Ollivier-Ricci curvature bounds in hypergraphs, bridging theoretical insights with scalable computation for large networks.

Contribution

It extends Ollivier-Ricci curvature bounds to hypergraphs and proposes a simplified, efficient computational approach suitable for large-scale graph analysis.

Findings

Method achieves linear computational complexity.

Significant improvements demonstrated on synthetic datasets.

Effective application to real-world large networks.

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.