Exactly scale-free scale-free networks

TL;DR

This paper introduces a maximum entropy process to characterize typical properties of scale-free networks, revealing that preferential attachment models produce atypical, hub-centric structures and providing tools to assess real networks' typicality.

Contribution

It proposes a maximum entropy framework for understanding typical scale-free network properties and introduces a method to compare real networks against this standard.

Findings

Preferential attachment networks are not representative of typical scale-free networks.

The hub-centric structure of preferential attachment models causes their atypical fragility.

A new surrogate method allows statistical testing of real networks' typicality.

Abstract

Many complex natural and physical systems exhibit patterns of interconnection that conform, approximately, to a network structure referred to as scale-free. Preferential attachment is one of many algorithms that have been introduced to model the growth and structure of scale-free networks. With so many different models of scale-free networks it is unclear what properties of scale-free networks are typical, and what properties are peculiarities of a particular growth or construction process. We propose a simple maximum entropy process which provides the best representation of what are typical properties of scale-free networks, and provides a standard against which real and algorithmically generated networks can be compared. As an example we consider preferential attachment and find that this particular growth model does not yield typical realizations of scale-free networks. In particular, the widely discussed "fragility" of scale-free networks is actually found to be due to the peculiar "hub-centric" structure of preferential attachment networks. We provide a method to generate or remove this latent hub-centric bias --- thereby demonstrating exactly which features of preferential attachment networks are atypical of the broader class of scale-free networks. We are also able to statistically demonstrate whether real networks are typical realizations of scale-free networks, or networks with that particular degree distribution; using a new surrogate generation method for complex networks, exactly analogous the the widely used surrogate tests of nonlinear time series analysis.