Diameter of the stochastic mean-field model of distance

TL;DR

This paper extends the understanding of the maximum distance in a stochastic mean-field model, showing it converges in distribution to a limit related to an infinite random structure, connecting it to Erdős-Rényi graph diameter results.

Contribution

It proves the distributional convergence of the re-centered maximum distance in the mean-field model, linking it to a limiting infinite random structure and previous Erdős-Rényi graph findings.

Findings

Maximum distance scaled by log n converges in distribution after re-centering.

Limiting distribution identified via a maximization on an infinite random structure.

Connection established between the model's diameter and Erdős-Rényi graph diameter limits.

Abstract

We consider the complete graph $\cK_n$ on $n$ vertices with exponential mean $n$ edge lengths. Writing $C_{ij}$ for the weight of the smallest-weight path between vertex $i,j\in [n]$, Janson showed that $\max_{i,j\in [n]} C_{ij}/\log{n}$ converges in probability to 3. We extend this result by showing that $\max_{i,j\in [n]} C_{ij} - 3\log{n}$ converges in distribution to a limiting random variable that can be identified via a maximization procedure on a limiting infinite random structure. Interestingly, this limiting random variable has also appeared as the weak limit of the re-centered graph diameter of the barely supercritical Erd\H{o}s-R\'enyi random graph in work by Riordan and Wormald.