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.
Highlights
Curvature is one of the most basic geometric characteristics of space
We show that Ollivier curvature [43,44,45] of random geometric graphs [46,47,48] in any Riemannian manifold converges to Ricci curvature of the manifold in the continuum limit
We describe our generalization of the Ollivier curvature definition to mesoscopic neighborhoods in graphs in general and in random geometric graphs in particular, Sec
Summary
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 extensions of curvature definitions applicable to graphs, simplicial complexes, and other discrete structures. Another class of approaches aimed at explaining spacetime emergence from “quantum bits” appears in space-from-entanglement proposals such as ER=EPR or tensor networks [34,35,36,37,38,39,40] In all these examples, one can hope that a particular discrete structure represents Planck-scale gravitational physics, only if discrete curvature of this structure converges to Ricci curvature of classical spacetime in the continuum limit. We perform large-scale simulations that agree with our proofs and suggest that the convergence windows are much wider than the limits imposed by our proof techniques These results link rigorously a definition of graph curvature to the traditional Ricci curvature of a Riemannian manifold.
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