A Bound for the Diameter of Random Hyperbolic Graphs

Abstract

Random hyperbolic graphs were recently introduced by Krioukov et. al. [KPK+10] 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 > ½, C ∊ ℝ, n ∊ ℕ, set R = 2lnn + C and build the graph G = (V,E) with |V| = n as follows: For each ν ∊ V, generate i.i.d. polar coordinates (rν,θν) using the joint density function f(r,θ), with θv chosen uniformly from [0, 2π) and rν with density 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 ½ < α < 1 a.a.s. for any two vertices of the same component, their graph distance is O(logCo+1+o(1)), where , 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(log2Co+1+o(1)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(log n), thus yielding a lower bound on the size of the second largest component.