On the spread of random graphs

TL;DR

This paper investigates the spread of various sparse random graph models, establishing bounds and asymptotic behaviors, including for supercritical Erdős–Rényi graphs, random regular graphs, and small world models.

Contribution

It provides new bounds and asymptotic results for the spread of several classes of sparse random graphs, extending understanding of their connectivity properties.

Findings

Spread of supercritical G_{n,p} is bounded with high probability

Spread of G(n,d) approaches that of complete graphs as d increases

Lower bounds on spread in barely supercritical regimes

Abstract

The spread of a connected graph G was introduced by Alon, Boppana and Spencer (1998) and measures how tightly connected the graph is. It is defined as the maximum over all Lipschitz functions f on V(G) of the variance of f(X) when X is uniformly distributed on V(G). We investigate the spread for certain models of sparse random graph; in particular for random regular graphs G(n,d), for Erd\H{o}s-R\'enyi random graphs G_{n,p} in the supercritical range p>1/n, and for a 'small world' model. For supercritical G_{n,p}, we show that if p=c/n with c>1 fixed then with high probability the spread of the giant component is bounded, and we prove corresponding statements for other models of random graphs, including a model with random edge-lengths. We also give lower bounds on the spread for the barely supercritical case when p=(1+o(1))/n. Further, we show that for d large, with high probability the spread of G(n,d) becomes arbitrarily close to that of the complete graph K_n.