The Scaling Window for a Random Graph with a Given Degree Sequence.

We consider a random graph on a given degree sequence D, satisfying certain conditions. Molloy and Reed defined a parameter Q = Q(D) and proved that Q = 0 is the threshold for the random graph to have a giant component. We introduce a new parameter R = R( \documentclass{article}\usepackage{mathrsfs, amsmath, amssymb}\pagestyle{empty}\begin{document}\begin{align*}\mathcal {D}\end{align*} \end{document}) and prove that if |Q| = O(n-1/3R2/3) then, with high probability, the size of the largest component of the random graph will be of order T(n2/3R-1/3). If |Q| is asymptotically larger than n-1/3R2/3 then the size of the largest component is asymptotically smaller or larger than n2/3R-1/3. Thus, we establish that the scaling window is |Q| = O(n-1/3R2/3). (C) 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2012