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Register a new userOllivier curvature of random geometric graphs converges to Ricci curvature of their Riemannian manifolds
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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 lurking and moot. In particular, it is not rigorously known if any notion of graph curvature converges to any traditional notion of curvature of smooth space. Here we prove that in proper settings the Ollivier-Ricci curvature of random geometric graphs in Riemannian manifolds converges to the Ricci curvature of the manifold. This is the first rigorous result linking curvature of random graphs to curvature of smooth spaces. Our results hold for different notions of graph distances, including the rescaled shortest path distance, and for different graph densities. With the scaling of the average degree, as a function of the graph size, ranging from nearly logarithmic to nearly linear.
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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.
We investigate some topological properties of random geometric complexes and random geometric graphs on Riemannian manifolds in the thermodynamic limit. In particular, for random geometric complexes we prove that the normalized counting measure of connected components, counted according to isotopy type, converges in probability to a deterministic measure. More generally, we also prove similar convergence results for the counting measure of types of components of each $k$-skeleton of a random geometric complex. As a consequence, in the case of the $1$-skeleton (i.e. for random geometric graphs) we show that the empirical spectral measure associated to the normalized Laplace operator converges to a deterministic measure.
We are concerned with the study of different notions of curvature on graphs. We show that if a graph has stronger inner-outer curvature growth than a model graph, then it has faster volume growth too. We also study the relationhips of volume growth with other kind of curvatures, such as the Ollivier-Ricci curvature.
We investigate the topologies of random geometric complexes built over random points sampled on Riemannian manifolds in the so-called thermodynamic regime. We prove the existence of universal limit laws for the topologies; namely, the random normalized counting measure of connected components (counted according to homotopy type) is shown to converge in probability to a deterministic probability measure. Moreover, we show that the support of the deterministic limiting measure equals the set of all homotopy types for Euclidean geometric complexes of the same dimension as the manifold.
We examine questions of geometric realizability for algebraic structures which arise naturally in affine and Riemannian geometry. Suppose given an algebraic curvature operator R at a point P of a manifold M and suppose given a real analytic (resp. C-k for finite k at least 2) pseudo-Riemannian metric on M defined near P. We construct a torsion free real analytic (resp. C-k) connection D which is defined near P on the tangent bundle of M whose curvature operator is the given operator R at P and so that D has constant scalar curvature. We show that if R is Ricci symmetric, then D can be chosen to be Ricci symmetric; if R has trace free Ricci tensor, then D can be chosen to have trace free Ricci tensor; if R is Ricci alternating, then D can be chosen to be Ricci alternating.
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