A critical point for random graphs with a given degree sequence

Abstract
Given a sequence of nonnegative real numbers λ0, λ1… which sum to 1, we consider random graphs having approximately λi n vertices of degree i. Essentially, we show that if Σ i(i ‐ 2)λi > 0, then such graphs almost surely have a giant component, while if Σ i(i 2)λ. < 0, then almost surely all components in such graphs are small. We can apply these results to Gn,p,Gn.M, and other well‐known models of random graphs. There are also applications related to the chromatic number of sparse random graphs.

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