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Limit theory of sparse random geometric graphs in high dimensions

Author

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  • Bonnet, Gilles
  • Hirsch, Christian
  • Rosen, Daniel
  • Willhalm, Daniel

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.

Suggested Citation

  • Bonnet, Gilles & Hirsch, Christian & Rosen, Daniel & Willhalm, Daniel, 2023. "Limit theory of sparse random geometric graphs in high dimensions," Stochastic Processes and their Applications, Elsevier, vol. 163(C), pages 203-236.
    • Handle: RePEc:eee:spapps:v:163:y:2023:i:c:p:203-236
    DOI: 10.1016/j.spa.2023.06.002
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    References listed on IDEAS

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    1. Owada, Takashi, 2019. "Topological crackle of heavy-tailed moving average processes," Stochastic Processes and their Applications, Elsevier, vol. 129(12), pages 4965-4997.
    2. Johannes Krebs & Christian Hirsch, 2022. "Functional central limit theorems for persistent Betti numbers on cylindrical networks," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 49(1), pages 427-454, March.
    3. Grygierek, Jens & Thäle, Christoph, 2020. "Gaussian fluctuations for edge counts in high-dimensional random geometric graphs," Statistics & Probability Letters, Elsevier, vol. 158(C).
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