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arXiv · 1504.00754

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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Abstract

We present two complementary analytical approaches for calculating the distribution of shortest path lengths in Erdos-R\'enyi 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 that the mean degree scales as $N^{\alpha}$ with the network size, the distribution becomes extremely narrow in the asymptotic limit, namely almost all pairs of nodes are equidistant, at distance $d=\lfloor 1/\alpha \rfloor$ 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.

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Eytan Katzav, Mor Nitzan, Daniel ben-Avraham, P. L. Krapivsky, Reimer Kühn, Nathan Ross, Ofer Biham. 2015-04-03. Analytical results for the distribution of shortest path lengths in random networks. https://doi.org/10.1209/0295-5075/111/26006

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