Distance Distribution between Complex Network Nodes in Hyperbolic Space

In the emerging field of network science, a recent model proposes that a hyperbolic geometry underlies the network representation of complex systems, shaping their topology and being responsible for their signature features: scale invariance and strong clustering. Under this model of network formation, points representing system components are placed in a hyperbolic circle and connected if the distance between them is below a certain threshold. Then the aforementioned properties come out naturally, as a direct consequence of the geometric principles of the hyperbolic space containing the network. With the aim of providing insights into the stochastic processes behind the structure of complex networks constructed with this model, the probability density for the approximate hyperbolic distance between N points, distributed quasi-uniformly at random in a disk of radius R ~ lnN, is determined in this paper, together with other density functions needed to derive this result.