Ollivier-Ricci curvature convergence in random geometric graphs

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 known converge in any limit traditional definition curvature a Riemannian manifold. Here we show that Ollivier random geometric graphs manifold converges the continuum Ricci underlying manifold, but only if properly generalized apply mesoscopic neighborhoods. This result establishes first rigorous link applicable networks smooth spaces.