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2020 Limit theory for isolated and extreme points in hyperbolic random geometric graphs
Nikolaos Fountoulakis, Joseph Yukich
Electron. J. Probab. 25: 1-51 (2020). DOI: 10.1214/20-EJP531


Given $\alpha \in (0, \infty )$ and $r \in (0, \infty )$, let ${\mathcal {D}}_{r, \alpha }$ be the disc of radius $r$ in the hyperbolic plane having curvature $-\alpha ^{2}$. Consider the Poisson point process having uniform intensity density on ${\mathcal {D}}_{R, \alpha }$, with $R = 2 \log (n/ \nu )$, $n \in \mathbb {N}$, and $\nu < n$ a fixed constant. The points are projected onto ${\mathcal {D}}_{R, 1}$, preserving polar coordinates, yielding a Poisson point process ${\mathcal {P}}_{\alpha , n}$ on ${\mathcal {D}}_{R, 1}$. The hyperbolic geometric graph ${\mathcal {G}}_{\alpha , n}$ on ${\mathcal {P}}_{\alpha , n}$ puts an edge between pairs of points of ${\mathcal {P}}_{\alpha , n}$ which are distant at most $R$. This model has been used to express fundamental features of complex networks in terms of an underlying hyperbolic geometry.

For $\alpha \in (1/2, \infty )$ we establish expectation and variance asymptotics as well as asymptotic normality for the number of isolated and extreme points in ${\mathcal {G}}_{\alpha , n}$ as $n \to \infty $. The limit theory and renormalization for the number of isolated points are highly sensitive on the curvature parameter. In particular, for $\alpha \in (1/2, 1)$, the variance is super-linear, for $\alpha = 1$ the variance is linear with a logarithmic correction, whereas for $\alpha \in (1, \infty )$ the variance is linear. The central limit theorem fails for $\alpha \in (1/2, 1)$ but it holds for $\alpha \in (1, \infty )$.


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Nikolaos Fountoulakis. Joseph Yukich. "Limit theory for isolated and extreme points in hyperbolic random geometric graphs." Electron. J. Probab. 25 1 - 51, 2020.


Received: 21 June 2019; Accepted: 3 October 2020; Published: 2020
First available in Project Euclid: 10 December 2020

MathSciNet: MR4186260
Digital Object Identifier: 10.1214/20-EJP531

Primary: 05C80
Secondary: 05C12 , 05C82

Keywords: central limit theorem , complex networks , hyperbolic plane , Random geometric graphs


Vol.25 • 2020
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