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

Abstract

We consider a random graph on a given degree sequence D, satisfying certain conditions. We focus on two parameters Q = Q(D),R = R(D). Molloy and Reed proved that Q = 0 is the threshold for the random graph to have a giant component. We 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 ⊖(n2/3R-1/3). If Q is asymptotically larger/smaller that n-1/3R2/3 then the size of the largest component is asymptotically larger/smaller than n2/3R -1/3. In other words, we establish that |Q| = O(n -1/3R2/3) is the scaling window. Copyright © by SIAM.