On the geometrisation of the Chung-Lu model and its component structure

We consider a model for complex networks that was introduced by Krioukov et al. [18], in which conceptually the intrinsic hierarchies of a network are mapped into the hyperbolic plane. This model gives rise to a random graph on the hyperbolic plane, which turns out to behave locally like the well-known Chung-Lu model. The latter is a special case of inhomogeneous random graphs, where two nodes are joined independently with probability proportional to the product of some pre-assigned weights whose distribution follows a power law. However, in our setting independence is no longer present. In fact, it is the existence of dependencies that gives rise to clustering, which is a ubiquitous property of complex networks. More specifically, in our setting, N points are chosen randomly on the hyperbolic plane and any two of them are joined by an edge if they are within a certain hyperbolic distance. The N points are distributed according to a quasi-uniform distribution, which is a distorted version of the uniform distribution and is controlled by a parameter α – when α = 1 this coincides with the uniform distribution. The present paper focuses on the evolution of the component structure of the random graph as this is determined by α. For α > 1, we show that with high probability as N grows the largest component of the random graph has sublinear order. When α crosses 1 a “giant” component of linear order emerges. ∗This research has been supported by a Marie Curie Career Integration Grant PCIG09-GA2011-293619.