TY - JOUR
T1 - Limit theory of sparse random geometric graphs in high dimensions
AU - Bonnet, Gilles
AU - Hirsch, Christian
AU - Rosen, Daniel
AU - Willhalm, Daniel
N1 - Funding Information:
The authors thank D. Yogeshwaran for pointing out the relation between Poisson approximation and CLT elucidated in Remark 5.4 . The authors acknowledge financial support of the CogniGron research center, Netherlands and the Ubbo Emmius Funds (University of Groningen), Netherlands .
Publisher Copyright:
© 2023 The Author(s)
PY - 2023/9
Y1 - 2023/9
N2 - 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.
AB - 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.
KW - Betti numbers
KW - Functional central limit theorem
KW - High dimension
KW - Poisson approximation
KW - Random geometric graph
UR - http://www.scopus.com/inward/record.url?scp=85163357648&partnerID=8YFLogxK
UR - https://arxiv.org/pdf/2212.12268
U2 - 10.1016/j.spa.2023.06.002
DO - 10.1016/j.spa.2023.06.002
M3 - Article
AN - SCOPUS:85163357648
SN - 0304-4149
VL - 163
SP - 203
EP - 236
JO - Stochastic processes and their applications
JF - Stochastic processes and their applications
ER -