On the power-law random graph model of massive data networks

Simplifying stochastic models of the topology of Internet have been studied intensively during the past few years. One of the most interesting ones is a random graph, where the degrees of the N nodes are drawn independently from a distribution with a Pareto tail with index τ∈(2,3) (finite mean and infinite variance), and the connections are then made randomly. We show that, asymptotically almost surely, the graph has a giant component, and the distance between two randomly selected nodes of the giant component is of the order log log N. This high connectivity is a consequence of the spontaneous emergence of a “core network” consisting of nodes with high degrees. Our result sheds light on the structure of the random graph model and raises interesting issues on its similarities and dissimilarities with the real Internet