2018/02/16 by Takashi Owada, Owada, Takashi, D. Yogeshwaran +1
Mathematics · #05C80 #51M10 #60D05 #60F05 #Combinatorics (math.CO) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric Topology (math.GT) #Geometry and complex manifolds #Mathematical Dynamics and Fractals #Probability (math.PR)
paper · pdf · doi:10.48550/arxiv.1802.06105
openalex publication_date 2018/02/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the hyperbolic random geometric graph introduced in Krioukov et al. For a sequence Rn → ∞, we define these graphs to have the vertex set as Poisson points distributed uniformly in balls B(0,Rn) ⊂ Bdα, the d-dimensional Poincaré ball (unit d-ball with the Poincaré metric dα corresponding to negative curvature -α2, α> 0) by connecting any two points within a distance Rn according to the metric dζ, ζ> 0. Denoting these graphs by HGn(Rn ; α, ζ), we study asymptotic counts of copies of a fixed tree Γk (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)]-1log (n/ ν), ν∈ (0,∞). The latter choice of Rn for α/ ζ> 1/2 corresponds to the thermodynamic regime. We show multiple phase transitions in HGn(Rn ; α, ζ) as α/ ζ increases, i.e., the space Bdα becomes 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 Γk 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 Last et al. derived using the Malliavin-Stein method.