  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.

Probability;Disordered Systems and Neural Networks;Social and Information Networks;General Relativity and Quantum Cosmology;Physics and Society