Description:

Curvature is one of the most basic geometric characteristics of space. The original definitions of curvature apply only to smooth Riemannian or Lorentzian manifolds, but there exist numerous nonequivalent extensions of these definitions applicable to graphs and simplicial complexes. Unfortunately, no notion of graph curvature is known to converge in any limit to any curvature of any smooth space. We show that the Ollivier curvature of random geometric graphs in any Riemannian manifold converges in the continuum limit to the Ricci curvature of the manifold. Random geometric graphs are fundamental in topology since they are 1-skeletons of Rips complexes whose topology is known to converge to the manifold topology. In that context, we show that their geometry also converges to the manifold geometry. This result establishes the first rigorous connection between curvatures of random discrete objects and smooth continuous spaces, proving correct original Riemann's ideas behind the foundations of Riemannian geometry, with applications ranging from traditional tasks in network science and machine learning (graph embedding, manifold learning, etc) to foundational problems in nonperturbative quantum gravity.

Professor Dmitri Krioukov, Northeastern University

Host: István Kovács

Keywords: Physics, Astronomy, Complex Systems