A new approach to the giant component problem

We study the largest component of a random (multi)graph on n vertices with a given degree sequence. We let n →∞. 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 [24,25] 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+ω(n)n-1/3, where &omega(n)→∞ arbitrarily slowly. Our method is based on the properties of empirical distributions of independent random variables, and leads to simple proofs.