An old approach to the giant component problem

TL;DR

This paper revisits a classical result on the size of the giant component in random graphs, providing a stronger version with exponential bounds under minimal assumptions.

Contribution

It strengthens Molloy and Reed's 1998 result by establishing exponential deviation bounds with minimal conditions.

Findings

Proves a strong form of the giant component size result

Provides exponential bounds on large deviations

Applies under minimal conditions

Abstract

In 1998, Molloy and Reed showed that, under suitable conditions, if a sequence of degree sequences converges to a probability distribution $D$, then the size of the largest component in corresponding $n$-vertex random graph is asymptotically $\rho(D)n$, where $\rho(D)$ is a constant defined by the solution to certain equations that can be interpreted as the survival probability of a branching process associated to $D$. There have been a number of papers strengthening this result in various ways; here we prove a strong form of the result (with exponential bounds on the probability of large deviations) under minimal conditions.