Average path length in uncorrelated random networks with hidden variables

TL;DR

This paper derives an analytic formula for the average path length in various uncorrelated random networks with hidden variables, revealing saturation effects in large scale-free networks with certain degree distributions.

Contribution

It provides a unified analytic approach to calculate average path length in uncorrelated random networks with hidden variables, including classical and scale-free networks.

Findings

Average path length saturates for large networks with degree exponent 2<α<3.

Structural properties of scale-free networks are more complex than previously thought.

The results apply to real-world systems modeled by such networks.

Abstract

Analytic solution for the average path length in a large class of uncorrelated random networks with hidden variables is found. We apply the approach to classical random graphs of Erdos and Renyi (ER), evolving networks introduced by Barabasi and Albert (BA) as well as random networks with asymptotic scale-free connectivity distributions characterized by an arbitrary scaling exponent $\alpha>2$. Our result for $2<\alpha<3$ shows that structural properties of asymptotic scale-free networks including numerous examples of real-world systems are even more intriguing then ultra-small world behavior noticed in pure scale-free structures and for large system sizes $N\to\infty$ there is a saturation effect for the average path length.