Average path length in random networks

Details Average path length in random networks Average path length in random networks

2002-12-10

arxiv.org/abs/cond-mat/0212230v3

Physics

Condensed Matter

Disordered Systems and Neural Networks

4 pages, 2 figures, changed content

Scientific paper

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 $\alpha>3$.

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