Average path length in random networks

Analytic solution for the average path length in a large class of random graphs is found. We apply the approach to classical random graphs of Erdös and Rényi (ER) and to scale-free networks of Barabási and Albert (BA). In both cases our results confirm previous observations: small world behavior in classical random graphs $l_ER ∼ \ln N$ and ultra small world effect characterizing scale-free BA networks $l_BA ∼ \ln N/\ln\ln N$. In the case of scale-free random graphs with power law degree distributions we observed the saturation of the average path length in the limit of $N\to∞$ for systems with the scaling exponent $2< α <3$ and the small-world behaviour for systems with $α>3$.