Curvature is a fundamental geometric characteristic of smooth spaces. In recent years different notions of curvature have been developed for combinatorial discrete objects such as graphs. However, the connections between such discrete notions of curvature and their smooth counterparts remain unclear. In particular, it is not rigorously known if any notion of graph curvature converges to any traditional notion of curvature of smooth spaces. In this talk I will address this challenge by considering Ollivier-Ricci curvature on random geometric graphs on smooth and complete Riemannian manifolds. I will identify proper settings for which the Ollivier-Ricci curvature converges to the Ricci curvature of the manifold. These results create a bridge between the discrete structure of graphs and continuous geometry underneath them. The results hold for different notions of graph distances, including the rescaled shortest path distance, and for different graph densities, with the average degree ranging from nearly logarithmic to nearly linear in the graph size. I will also show numerical experiments on Euclidean, Spherical and Hyperbolic spaces which highlight the power of this discrete curvature and hint to some more interesting properties.
This is joint work with William Cunningham, Dmitri Krioukov, Gabor Lippner and Carlo Trugenberger