Sub-tree counts on hyperbolic random geometric graphs

Abstract

We study the hyperbolic random geometric graph introduced in Krioukov et al. For a sequence $R_n \to \infty$, we define these graphs to have the vertex set as Poisson points distributed uniformly in balls $B(0,R_n) \subset B_d^{\alpha}$, the $d$-dimensional Poincar\'e ball (unit d-ball with the Poincar\'e metric $d_{\alpha}$ corresponding to negative curvature $-\alpha^2, \alpha > 0$) by connecting any two points within a distance $R_n$ according to the metric $d_{\zeta}, \zeta > 0$. Denoting these graphs by $HG_n(R_n ; \alpha, \zeta)$, we study asymptotic counts of copies of a fixed tree $\Gamma_k$ (with the ordered degree sequence $d_{(1)} \leq \ldots \leq d_{(k)}$) in $HG_n(R_n ; \alpha, \zeta)$. Unlike earlier works, we count more involved structures, allowing for $d > 2$, and in many places, more general choices of $R_n$ rather than $R_n = 2[\zeta (d-1)]^{-1}\log (n/ \nu), \nu \in (0,\infty)$. The latter choice of $R_n$ for $\alpha / \zeta > 1/2$ corresponds to the thermodynamic regime. We show multiple phase transitions in $HG_n(R_n ; \alpha, \zeta)$ as $\alpha / \zeta$ increases, i.e., the space $B_d^{\alpha}$ becomes more hyperbolic. In particular, our analyses reveal that the sub-tree counts exhibit an intricate dependence on the degree sequence $d_{(1)},\ldots,d_{(k)}$ of $\Gamma_k$ as well as the ratio $\alpha/\zeta$. Under a more general radius regime $R_n$ than that described above, we investigate the asymptotics of the expectation and variance of sub-tree counts. Moreover, we prove the corresponding central limit theorem as well. Our proofs rely crucially on a careful analysis of the sub-tree counts near the boundary using Palm calculus for Poisson point processes along with estimates for the hyperbolic metric and measure. For the central limit theorem, we use the abstract normal approximation result from Last et al. derived using the Malliavin-Stein method.