Scale-free networks with a large- to hypersmall-world transition

Recently there has been a tremendous interest in models of networks with a power-law distribution of degree—so-called “scale-free networks.” It has been observed that such networks, normally, have extremely short path-lengths, scaling logarithmically or slower with system size. As an exotic and counterintuitive example we propose a simple stochastic model capable of generating scale-free networks with linearly scaling distances. Furthermore, by tuning a parameter the model undergoes a phase transition to a regime with extremely short average distances, apparently slower than loglogN (which we call a hypersmall-world regime). We characterize the degree–degree correlation and clustering properties of this class of networks.