Distribution of diameters for Erd\"os-R\'enyi random graphs

TL;DR

This paper investigates the distribution of diameters in Erd"os-Rényi graphs across different regimes, employing large-deviations techniques to analyze probabilities as small as 10^{-100} and revealing structural changes at certain connectivity thresholds.

Contribution

It provides the first comprehensive numerical analysis of the diameter distribution in Erd"os-Rényi graphs, including the full distribution for large deviations and the structural transition at c=1.

Findings

Distribution matches analytical results for c<1.

Distribution exhibits an inflection point for c>1.

Finite-size rate function suggests large deviation principle holds.

Abstract

We study the distribution of diameters d of Erd\"os-R\'enyi random graphs with average connectivity c. The diameter d is the maximum among all shortest distances between pairs of nodes in a graph and an important quantity for all dynamic processes taking place on graphs. Here we study the distribution P(d) numerically for various values of c, in the non-percolating and the percolating regime. Using large-deviations techniques, we are able to reach small probabilities like 10^{-100} which allow us to obtain the distribution over basically the full range of the support, for graphs up to N=1000 nodes. For values c<1, our results are in good agreement with analytical results, proving the reliability of our numerical approach. For c>1 the distribution is more complex and no complete analytical results are available. For this parameter range, P(d) exhibits an inflection point, which we found to be related to a structural change of the graphs. For all values of c, we determined the finite-size rate function Phi(d/N) and were able to extrapolate numerically to N->infinity, indicating that the large deviation principle holds.