The scaling window for a random graph with a given degree sequence

Abstract

Abstract 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 ‐1/3 R 2/3 ) then, with high probability, the size of the largest component of the random graph will be of order Θ( n 2/3 R ‐1/3 ). If | Q | is asymptotically larger than n ‐1/3 R 2/3 then the size of the largest component is asymptotically smaller or larger than n 2/3 R ‐1/3 . Thus, we establish that the scaling window is | Q | = O ( n ‐1/3 R 2/3 ). © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2012