Exact and Asymptotic Results on Coarse Ricci Curvature of Graphs

TL;DR

This paper provides exact and asymptotic bounds for Ollivier's Ricci curvature on various classes of graphs, including bipartite, girth-specific, and random graphs, advancing understanding of graph curvature properties.

Contribution

It introduces new bounds and characterizations for Ricci curvature on graphs, including girth-specific and regular graphs, and analyzes asymptotic behavior on random graph models.

Findings

Upper bounds for bipartite and girth ≥ 5 graphs' Ricci curvature

Characterization of Ricci-flat graphs of girth 5

Asymptotic Ricci curvature results for random bipartite and general graphs

Abstract

Ricci curvature was proposed by Ollivier in a general framework of metric measure spaces, and it has been studied extensively in the context of graphs in recent years. In this paper we prove upper bounds for Ollivier's Ricci curvature for bipartite graphs and for the graphs with girth at least 5. We also prove a general lower bound on the Ricci curvature in terms of the size of the maximum matching in an appropriate subgraph. As a consequence, we characterize the Ricci-flat graphs of girth 5. Moreover, using our general lower bound and the Birkhoff-von Neumann theorem, we give a necessary and sufficient condition for the structure of Ricci-flat regular graphs of girth 4. Finally, we obtain the asymptotic Ricci curvature of random bipartite graphs $G(n,n, p)$ and random graphs $G(n, p)$, in various regimes of $p$.