Universal diameter bounds for random graphs with given degrees

TL;DR

This paper establishes universal bounds on the diameter of random graphs with prescribed degrees, showing that typically the diameter is at most proportional to the square root of the number of vertices, unless many vertices have degree 2.

Contribution

It proves that for most degree sequences, the expected diameter of a random graph is O(√n), and demonstrates that graphs with minimum degree 3 are highly connected with logarithmic diameter.

Findings

Expected diameter is O(√n) unless many vertices have degree 2.

Graphs with minimum degree ≥ 3 are connected with high probability and have logarithmic diameter.

Bounds are optimal for general degree sequences and trees.

Abstract

Given a graph $G$, let $\mathrm{diam}(G)$ be the greatest distance between any two vertices of $G$ which lie in the same connected component, and let $\mathrm{diam}^+(G)$ be the greatest distance between any two vertices of $G$; so $\mathrm{diam}^+(G)=\infty$ if $G$ is not connected. Fix a sequence $(d_1,\ldots,d_n)$ of positive integers, and let $\mathbf{G}$ be a uniformly random connected simple graph with $V(\mathbf{G})=[n]:=\{1,\ldots,n\}$ such that $\mathrm{deg}_{\mathbf{G}}(v)=d_v$ for all $v \in [n]$. We show that, unless a $1-o(1)$ proportion of vertices have degree $2$, then $\mathbb{E}[\mathrm{diam}(\mathbf{G})]=O(\sqrt{n})$. It is not hard to see that this bound is best possible for general degree sequences (and in particular in the case of trees, in which $\sum_{v=1}^n d_v = 2(n-1)$). We also prove that this bound holds without the connectivity constraint. As a key input to the proofs, we show that graphs with minimum degree $3$ are with high probability connected and have logarithmic diameter: if $\min(d_1,\ldots,d_n) \ge 3$ and $\mathbf{G}$ is a uniformly random simple graph with $V(\mathbf{G})=[n]$ such that $\mathrm{deg}_{\mathbf{G}}(v)=d_v$ for all $v \in [n]$, then $\mathrm{diam}^+(\mathbf{G})=$ $O_{\mathbb{P}}(\log n)$; this bound is also best possible.