Metric structure of random networks

TL;DR

This paper develops a rigorous method to analyze shortest path statistics in random networks with arbitrary degree distributions, accounting for loops and revealing finite width in distance distributions.

Contribution

It introduces a novel approach that extends beyond tree approximations to accurately include loops, providing detailed distance distribution analysis.

Findings

Distance distribution has finite width in large networks.

Mean intervertex distance grows with network size.

Method applies to various degree distributions.

Abstract

We propose a consistent approach to the statistics of the shortest paths in random graphs with a given degree distribution. This approach goes further than a usual tree ansatz and rigorously accounts for loops in a network. We calculate the distribution of shortest-path lengths (intervertex distances) in these networks and a number of related characteristics for the networks with various degree distributions. We show that in the large network limit this extremely narrow intervertex distance distribution has a finite width while the mean intervertex distance grows with the size of a network. The size dependence of the mean intervertex distance is discussed in various situations.