We consider a random graph on a given degree sequence @math , satisfying certain conditions. We focus on two parameters @math . Molloy and Reed proved that Q=0 is the threshold for the random graph to have a giant component. We prove that if @math then, with high probability, the size of the largest component of the random graph will be of order @math . If @math is asymptotically larger than @math then the size of the largest component is asymptotically smaller or larger than @math . Thus, we establish that the scaling window is @math .
We consider random graphs with a fixed degree sequence. Molloy and Reed [11, 12] studied how the size of the giant component changes according to degree conditions. They showed that there is a phase transition and investigated the order of components before and after the critical phase. In this paper we study more closely the order of components at the critical phase, using singularity analysis of a generating function for a branching process which models the random graph with a given degree sequence. 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. © 1995 Wiley Periodicals, Inc.