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 .
Kang and Seierstad @cite_4 applied generating functions to study the case where @math , but is outside of the scaling window. They require a maximum degree of at most @math and that the degree sequences satisfy certain conditions that are stronger than those in @cite_12 ; one of these conditions implies that @math is bounded by a constant. Based on what is known for @math , it was natural to guess that for @math we would have @math . They proved that if @math then @math , and if @math then @math . So for the case where @math is bounded, this almost confirmed that natural guess - except that they did not cover the range where @math .
@cite_4 @cite_12
2168564667 2044881936
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.