Average path length in random networks

TL;DR

This paper derives an analytical expression for the average path length in various random networks, confirming known behaviors and revealing saturation effects in scale-free networks with certain degree exponents.

Contribution

It provides a unified analytical approach to compute average path length in different classes of random graphs, including Erdős-Rényi and scale-free networks, and identifies saturation phenomena.

Findings

Average path length in ER graphs scales as ln N.

Scale-free BA networks exhibit ultra small world behavior with ln N / ln ln N.

Saturation of average path length occurs for scale-free networks with 2<α<3.

Abstract

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 Erd\"{o}s and R\'{e}nyi (ER) and to scale-free networks of Barab\'{a}si and Albert (BA). In both cases our results confirm previous observations: small world behavior in classical random graphs $l_{ER} \sim \ln N$ and ultra small world effect characterizing scale-free BA networks $l_{BA} \sim \ln N/\ln\ln N$. 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 $N\to\infty$ for systems with the scaling exponent $2< \alpha <3$ and the small-world behaviour for systems with $\alpha>3$.