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Sub-tree counts on hyperbolic random geometric graphs

Part of: Limit theorems Real and complex geometry Graph theory Geometric probability and stochastic geometry

Published online by Cambridge University Press:  13 June 2022

Takashi Owada  and
D. Yogeshwaran Open the ORCID record for D. Yogeshwaran [Opens in a new window]
Takashi Owada*
Affiliation:
Purdue University
D. Yogeshwaran*
Affiliation:
Indian Statistical Institute
*
*Postal address: Department of Statistics, Purdue University, West Lafayette, IN 47907, USA. Email address: owada@purdue.edu
**Postal address: Theoretical Statistics and Mathematics Unit, Indian Statistical Institute, 8th Mile, Mysore Road, Bangalore, 560059, INDIA. Email address: d.yogesh@isibang.ac.in
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Abstract

The hyperbolic random geometric graph was introduced by Krioukov et al. (Phys. Rev. E82, 2010). Among many equivalent models for the hyperbolic space, we study the d-dimensional Poincaré ball ( $d\ge 2$ ), with a general connectivity radius. While many phase transitions are known for the expectation asymptotics of certain subgraph counts, very little is known about the second-order results. Two of the distinguishing characteristics of geometric graphs on the hyperbolic space are the presence of tree-like hierarchical structures and the power-law behaviour of the degree distribution. We aim to reveal such characteristics in detail by investigating the behaviour of sub-tree counts. We show multiple phase transitions for expectation and variance in the resulting hyperbolic geometric graph. In particular, the expectation and variance of the sub-tree counts exhibit an intricate dependence on the degree sequence of the tree under consideration. Additionally, unlike the thermodynamic regime of the Euclidean random geometric graph, the expectation and variance may exhibit different growth rates, which is indicative of power-law behaviour. Finally, we also prove a normal approximation for sub-tree counts using the Malliavin–Stein method of Last et al. (Prob. Theory Relat. Fields165, 2016), along with the Palm calculus for Poisson point processes.

Keywords

Hyperbolic spacesrandom geometric graphsPoisson point processsub-tree countscentral limit theoremMalliavin–Stein method

MSC classification

Primary: 60F05: Central limit and other weak theorems 60D05: Geometric probability and stochastic geometry
Secondary: 05C80: Random graphs 51M10: Hyperbolic and elliptic geometries (general) and generalizations
Type
Original Article
Information
Advances in Applied Probability , Volume 54 , Issue 4 , December 2022 , pp. 1032 - 1069
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

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