The probability of connectivity in a hyperbolic model of complex networks

We consider a model for complex networks that was introduced by Krioukov et al. (Phys Rev E 82 (2010) 036106). In this model, N points are chosen randomly inside a disk on the hyperbolic plane according to a distorted version of the uniform distribution and any two of them are joined by an edge if they are within a certain hyperbolic distance. This model exhibits a power-law degree sequence, small distances and high clustering. The model is controlled by two parameters α and ν where, roughly speaking, α controls the exponent of the power-law and ν controls the average degree. In this paper we focus on the probability that the graph is connected. We show the following results. For α > 1/2 and ν arbitrary, the graph is disconnected with high probability. For α < 1/2 and ν arbitrary, the graph is connected with high probability. When α = 1/2 and ν is fixed then the probability of being connected tends to a constant f(ν) that depends only on ν, in a continuous manner. Curiously, f(ν) = 1 for ν ≥ Π while it is strictly increasing, and in particular bounded away from zero and one, for 0 < ν < Π