On the evolution of random geometric graphs on spaces of negative curvature∗

We study a family of random geometric graphs on hyperbolic spaces. In this setting, N points are chosen randomly on the hyperbolic plane of curvature −ζ and any two of them are joined by an edge with probability that depends on their hyperbolic distance, independently of every other pair. In particular, when the positions of the points have been fixed, the distribution over the set of graphs on these points is the Boltzmann distribution, where the Hamiltonian is given by the sum of weighted indicator functions for each pair of points. The weight is proportional to a real parameter β > 0 (interpreted as the inverse temperature) as well as to the hyperbolic distance between the corresponding points. The N points are distributed according to a quasi-uniform distribution which is a distorted version of the uniform distribution and depends on a parameter α > 0 – when α = ζ it coincides with the uniform distribution. This class of random graphs was proposed by Krioukov et al. [17] as a model for complex networks. We provide a rigorous analysis of aspects of this model for all values of the ratio ζ/α, focusing on its dependence on the parameter β. We show that a phase transition occurs around β = 1. More specifically, let us assume that 0 < ζ/α < 2. In this case, we show that when β > 1 the degree of a typical vertex is bounded in probability (in fact it follows a distribution which for large values exhibits a power-law tail whose exponent depends only on ζ and α), whereas for β < 1 the degree is a random variable whose expected value grows polynomially in N . When β = 1, we establish logarithmic growth. For the case β > 1, we establish a connection with a class of inhomogeneous random graphs known as the Chung-Lu model. We show that the probability that two given points are adjacent is expressed through the kernel of this inhomogeneous random graph. We also consider the case ζ/α ≥ 2.