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

¹ Racah Institute of Physics, The Hebrew University - Jerusalem 91904, Israel
² Department of Microbiology and Molecular Genetics, Faculty of Medicine, The Hebrew University Jerusalem 91120, Israel
³ Department of Physics, Clarkson University - Potsdam, NY 13699-5820, USA
⁴ Department of Physics, Boston University - Boston, MA 02215, USA
⁵ Department of Mathematics, King's College London - Strand, London WC2R 2LS, UK
⁶ School of Mathematics and Statistics, University of Melbourne - Melbourne, VIC 3010, Australia

Received: 21 April 2015
Accepted: 20 July 2015

Abstract

We present two complementary analytical approaches for calculating the distribution of shortest path lengths in Erdős-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 with the network size, the distribution becomes extremely narrow in the asymptotic limit, namely almost all pairs of nodes are equidistant, at distance 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.