The diameter of sparse random graphs

TL;DR

This paper provides precise results on the diameter of sparse Erdős–Rényi random graphs across various regimes of edge probability, using branching process techniques to improve understanding of graph distances near phase transitions.

Contribution

It offers new, simplified proofs and refined bounds for the diameter of $G(n,p)$ in different regimes, completing the characterization of graph diameter with high accuracy.

Findings

Established $O_p(1)$ bounds for $p= heta/n$ with $ heta>1$

Proved 2-point concentration for $np oty$

Analyzed diameter behavior in the critical window $p=(1+epsilon)/n$

Abstract

In this paper we study the diameter of the random graph $G(n,p)$, i.e., the the largest finite distance between two vertices, for a wide range of functions $p=p(n)$. For $p=\la/n$ with $\la>1$ constant, we give a simple proof of an essentially best possible result, with an $O_p(1)$ additive correction term. Using similar techniques, we establish 2-point concentration in the case that $np\to\infty$. For $p=(1+\epsilon)/n$ with $\epsilon\to 0$, we obtain a corresponding result that applies all the way down to the scaling window of the phase transition, with an $O_p(1/\epsilon)$ additive correction term whose (appropriately scaled) limiting distribution we describe. Combined with earlier results, our new results complete the determination of the diameter of the random graph $G(n,p)$ to an accuracy of the order of its standard deviation (or better), for all functions $p=p(n)$. Throughout we use branching process methods, rather than the more common approach of separate analysis of the 2-core and the trees attached to it.