Limit theory of sparse random geometric graphs in high dimensions

Gilles Bonnet, Christian Hirsch, Daniel Rosen and Daniel Willhalm

Stochastic Processes and their Applications, 2023, vol. 163, issue C, 203-236

Abstract: We study topological and geometric functionals of l∞-random geometric graphs on the high-dimensional torus in a sparse regime, where the expected number of neighbors decays exponentially in the dimension. More precisely, we establish moment asymptotics, functional central limit theorems and Poisson approximation theorems for certain functionals that are additive under disjoint unions of graphs. For instance, this includes simplex counts and Betti numbers of the Rips complex, as well as general subgraph counts of the random geometric graph. We also present multi-additive extensions that cover the case of persistent Betti numbers of the Rips complex.

Keywords: Random geometric graph; High dimension; Functional central limit theorem; Poisson approximation; Betti numbers (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1016/j.spa.2023.06.002

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