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

TL;DR

This paper introduces a stochastic model for scale-free networks that exhibits a transition from large-world to hypersmall-world regimes, with extremely short path lengths and tunable properties.

Contribution

It presents a simple model capable of generating scale-free networks with both linear and ultra-short path lengths, revealing a phase transition in network topology.

Findings

Networks can have linearly scaling distances.

A phase transition to hypersmall-world regime is identified.

Degree correlation and clustering are characterized.

Abstract

Recently there have 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 en exotic and unintuitive 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 log log N (which we call a hypersmall-world regime). We characterize the degree-degree correlation and clustering properties of this class of networks.