An efficient alternative to Ollivier-Ricci curvature based on the Jaccard metric

TL;DR

This paper introduces two computationally efficient curvature measures based on the Jaccard coefficient as alternatives to Ollivier-Ricci curvature, demonstrating their effectiveness on various network models and real-world networks.

Contribution

The paper proposes two new Jaccard-based curvature metrics, JC and gJC, that are less computationally intensive and closely approximate Ollivier-Ricci curvature in large networks.

Findings

gJC closely matches Ollivier-Ricci curvature in large Erdos-Renyi graphs

gJC provides a good approximation for various network models and real networks

JC is effective only in specific scenarios

Abstract

We study Ollivier-Ricci curvature, a discrete version of Ricci curvature, which has gained popularity over the past several years and has found applications in diverse fields. However, the Ollivier-Ricci curvature requires an optimal mass transport problem to be solved, which can be computationally expensive for large networks. In view of this, we propose two alternative measures of curvature to Ollivier-Ricci which are motivated by the Jaccard coefficient and are demonstrably less computationally intensive, a cheaper Jaccard (JC) and a more expensive generalized Jaccard (gJC) curvature metric. We show theoretically that the gJC closely matches the Ollivier-Ricci curvature for Erdos-Renyi graphs in the asymptotic regime of large networks. Furthermore, we study the goodness of approximation between the proposed curvature metrics and Ollivier-Ricci curvature for several network models and real networks. Our results suggest that in comparison to an alternative curvature metric for graphs, the Forman-Ricci curvature, the gJC exhibits a reasonably good fit to the Ollivier-Ricci curvature for a wide range of networks, while the JC is shown to be a good proxy only for certain scenarios.