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

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 N^α 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.