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

TL;DR

This paper introduces two analytical methods to determine the distribution of shortest path lengths in Erdős-Rényi networks, revealing how the distribution narrows as network size increases and linking local degree to global distance properties.

Contribution

The paper develops two complementary analytical approaches for shortest path distribution in Erdős-Rényi networks, applicable to various network sizes and connectivities.

Findings

Distribution agrees with numerical simulations across network sizes.

As mean degree scales as N^α, the distribution becomes nearly degenerate.

Average shortest path length decreases with node degree m.

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.