A new approach to the giant component problem

TL;DR

This paper investigates the conditions under which a giant component emerges in large random graphs with prescribed degree sequences, providing new sharp thresholds and simplifying proofs using empirical distribution properties.

Contribution

It introduces a new approach to analyze the giant component problem, generalizing existing results and establishing sharp thresholds with simpler proofs.

Findings

Conditions for the emergence of a giant component

Sharp threshold results near the critical point

Simplified proofs using empirical distribution properties

Abstract

We study the largest component of a random (multi)graph on n vertices with a given degree sequence. We let n tend to infinity. Then, under some regularity conditions on the degree sequences, we give conditions on the asymptotic shape of the degree sequence that imply that with high probability all the components are small, and other conditions that imply that with high probability there is a giant component and the sizes of its vertex and edge sets satisfy a law of large numbers; under suitable assumptions these are the only two possibilities. In particular, we recover the results by Molloy and Reed on the size of the largest component in a random graph with a given degree sequence. We further obtain a new sharp result for the giant component just above the threshold, generalizing the case of G(n,p) with np=1+omega(n)n^{-1/3}, where omega(n) tends to infinity arbitrarily slowly. Our method is based on the properties of empirical distributions of independent random variables, and leads to simple proofs.