Ollivier-Ricci curvature convergence in random geometric graphs

TL;DR

This paper proves that Ollivier-Ricci curvature of random geometric graphs converges to the Ricci curvature of the underlying manifold when properly generalized, bridging discrete network curvature and continuous geometric curvature.

Contribution

It establishes the first rigorous connection between a network curvature definition and classical Riemannian curvature through a generalized Ollivier-Ricci curvature in the continuum limit.

Findings

Ollivier-Ricci curvature converges to Ricci curvature in the continuum limit.

Proper generalization of Ollivier curvature to mesoscopic neighborhoods is essential.

First rigorous link between network curvature and smooth space curvature.

Abstract

Connections between continuous and discrete worlds tend to be elusive. One example is curvature. Even though there exist numerous nonequivalent definitions of graph curvature, none is known to converge in any limit to any traditional definition of curvature of a Riemannian manifold. Here we show that Ollivier curvature of random geometric graphs in any Riemannian manifold converges in the continuum limit to Ricci curvature of the underlying manifold, but only if the definition of Ollivier graph curvature is properly generalized to apply to mesoscopic graph neighborhoods. This result establishes the first rigorous link between a definition of curvature applicable to networks and a traditional definition of curvature of smooth spaces.