Sub-tree counts on hyperbolic random geometric graphs

Abstract

In this article, we study the hyperbolic random geometric graph introduced recently in [28]. For a sequence Rn → ∞, we define these graphs to have the vertex set as Poisson points distributed uniformly in balls B(0, Rn) ⊂ B (α) d , the d-dimensional Poincaré ball (i.e., the unit ball on R d with the Poincaré metric dα corresponding to negative curvature −α 2 , α > 0) by connecting any two points within a distance Rn according to the metric d ζ , ζ > 0. Denoting these graphs by HGn(Rn; α, ζ), we study asymptotic counts of copies of a fixed tree Γ k (with the ordered degree sequence d (1) ≤ . . . ≤ d (k) ) in HGn(Rn; α, ζ). Unlike earlier works, we count more involved structures, allowing for d > 2, and in many places, more general choices of Rn rather than Rn = 2[ζ(d − 1)] −1 log(n/ν), ν ∈ (0, ∞). The latter choice of Rn for α/ζ > 1/2 corresponds to the thermodynamic regime in which the expected average degree is asymptotically constant. We show multiple phase transitions in HGn(Rn; α, ζ) as α/ζ increases, i.e., the space B (α) dbecomes more hyperbolic. In particular, our analyses reveal that the sub-tree counts exhibit an intricate dependence on the degree sequence d (1) , . . . , d (k) of Γ k as well as the ratio α/ζ. Under a more general radius regime Rn 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 [31] derived using the Malliavin-Stein method.

Intersections of Poisson $ k $-flats in constant curvature spaces

“…The aim of this branch of stochastic geometry is to distinguish those properties of a random geometric system which are universal to some extent from the ones which are sensitive to the underlying geometry, especially to the curvature of the underlying space. We mention by way of example the studies [6,7,20] on random convex hulls, the papers [5,21,22,26,27,29,30,32] on random tessellations as well as the works [4,8,15,16,17,18,40] on geometric random graphs and networks. The present paper continues this line of research and naturally connects to the articles [26,32].…”