On a geometrization of the Chung-Lu model for complex networks

We consider a model for complex networks that was introduced by Krioukov et al. [14], where the intrinsic hierarchies of a network are mapped into the hyperbolic plane. Krioukov et al. show that this model exhibits clustering and the distribution of its degrees has a power law tail. We show that asymptotically this model locally behaves like the well-known Chung-Lu model in which two nodes are joined independently with probability proportional to the product of some pre-assigned weights whose distribution follows a power law. Using this, we further determine exactly the asymptotic distribution of the degree of an arbitrary vertex. keywords: mathematical analysis of complex networks, hyperbolic geometry, degree distribution