Analytical results for the distribution of shortest path lengths in random networks

Eytan Katzav, Mor Nitzan, Daniel Ben-Avraham, P. L. Krapivsky, Reimer Kühn, Nathan Ross, Ofer Biham

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We present two complementary analytical approaches for calculating the distribution of shortest path lengths in Erdos-Rényi networks, based on recursion equations for the shells around a reference node and for the paths originating from it. The results are in agreement with numerical simulations for a broad range of network sizes and connectivities. The average and standard deviation of the distribution are also obtained. In the case in which the mean degree scales as Nα with the network size, the distribution becomes extremely narrow in the asymptotic limit, namely almost all pairs of nodes are equidistant, at distance d = [1/α] from each other. The distribution of shortest path lengths between nodes of degree m and the rest of the network is calculated. Its average is shown to be a monotonically decreasing function of m, providing an interesting relation between a local property and a global property of the network. The methodology presented here can be applied to more general classes of networks.

Original languageAmerican English
Article number26006
JournalLettere Al Nuovo Cimento
Issue number2
StatePublished - 1 Jul 2015

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