A Bound for the Diameter of Random Hyperbolic Graphs

Random hyperbolic graphs were recently introduced by Krioukov et. al. [KPK10] as a model for large networks. Gugelmann, Panagiotou, and Peter [GPP12] then initiated the rigorous study of random hyperbolic graphs using the following model: for α > 1 2 , C ∈ R, n ∈ N, set R = 2 lnn + C and build the graph G = (V,E) with |V | = n as follows: For each v ∈ V , generate i.i.d. polar coordinates (rv, θv) using the joint density function f(r, θ), with θv chosen uniformly from [0, 2π) and rv with density f(r) = α sinh(αr) cosh(αR)−1 for 0 ≤ r < R. Then, join two vertices by an edge, if their hyperbolic distance is at most R. We prove that in the range 1 2 < α < 1 a.a.s. for any two vertices of the same component, their graph distance is O(log0 n), where C0 = 2/( 1 2 − 3 4 α + α 2 4 ), thus answering a question raised in [GPP12] concerning the diameter of such random graphs. As a corollary from our proof we obtain that the second largest component has size O(log0 n), thus answering a question of Bode, Fountoulakis and Müller [BFM13]. We also show that a.a.s. there exist isolated components forming a path of length Ω(logn), thus yielding a lower bound on the size of the second largest component.