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 propose a random graph model which is a special case of sparse random graphs with given degree sequences. This model involves only a small number of parameters, called logsize and log-log growth rate. These parameters capture some universal characteristics of massive graphs. Furthermore, from these parameters, various properties of the graph can be derived. For example, for certain ranges of the parameters, we will compute the expected distribution of the sizes of the connected components which almost surely occur with high probability. We will illustrate the consistency of our model with the behavior of some massive graphs derived from data in telecommunications. We will also discuss the threshold function, the giant component, and the evolution of random graphs in this model.  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. A process for finishing keratinous material, especially rendering it shrink-resistant or imparting to it durably pressed effects, comprises 1. TREATING THE MATERIAL WITH A POLYTHIOL ESTER OF THE FORMULA [OH]q(s) ¦ R1-(CO)rO(CO)sR2SH]p ¦ [COOH]q(r) WHERE R1 represents an aliphatic or araliphatic hydrocarbon radical of at least 2 carbon atoms, which may contain not more than one ether oxygen atom, R2 represents a hydrocarbon radical, P IS AN INTEGER OF FROM 2 TO 6, Q IS ZERO OR A POSITIVE INTEGER OF AT MOST 3, SUCH THAT (P + Q) IS AT MOST 6, AND R AND S EACH REPRESENT ZERO OR 1 BUT ARE NOT THE SAME, AND 2. CURING THE POLYTHIOL ESTER ON THE MATERIAL BY MEANS OF A POLYENE CONTAINING, PER AVERAGE MOLECULE, AT LEAST TWO ETHYLENIC DOUBLE BONDS EACH beta TO AN OXYGEN, NITROGEN, OR SULFUR ATOM, THE SUM OF SUCH ETHYLENIC DOUBLE BONDS IN THE POLYENE AND OF THE MERCAPTAN GROUPS IN THE POLYTHIOL ESTER BEING MORE THAN 4 AND THE COMBINED WEIGHT OF THE POLYENE AND THE POLYTHIOL ESTER BEING FROM 0.5 TO 15 BY WEIGHT OF THE KERATINOUS MATERIAL TREATED.