Limit theory of isolated and extreme points in hyperbolic random geometric graphs

Given $\alpha \in (0, \infty)$ and $r \in (0, \infty)$, let ${\cal D}_{r, \alpha}$ be the disc of radius $r$ in the hyperbolic plane having curvature $-\alpha^2$. Consider the Poisson point process having uniform intensity density on ${\cal D}_{R, \alpha}$, with $R = 2 \log(n/ \nu),$ $n \in \mathbb{N}$, and $\nu < n$ a fixed constant. The points are projected onto ${\cal D}_{R, 1}$, preserving polar coordinates, yielding a Poisson point process ${\cal P}_{\alpha, n}$ on ${\cal D}_{R, 1}$. The hyperbolic geometric graph ${\cal G}_{\alpha, n}$ on ${\cal P}_{\alpha, n}$ puts an edge between pairs of points of ${\cal P}_{\alpha, n}$ 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 ${\cal G}_{\alpha, n}$ 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 $\alpha \in (1, \infty)$.