Average path length in random 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 Erd\"os and R\'enyi (ER), evolving networks introduced by Barab\'asi and Albert as well as random networks with asymptotic scale-free connectivity distributions characterized by an arbitrary scaling exponent $\ensuremath{\alpha}>2$. Our result for $2<\ensuremath{\alpha}<3$ shows that structural properties of asymptotic scale-free networks including numerous examples of real-world systems are even more intriguing than ultra-small world behavior noticed in pure scale-free structures and for large system sizes $N\ensuremath{\rightarrow}\ensuremath{\infty}$ there is a saturation effect for the average path length.