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

Abstract

We consider a random graph on a given degree sequence $\mathcal{D}$, satisfying certain conditions. We focus on two parameters $Q=Q(\mathcal{D}),R=R(\mathcal{D})$. Molloy and Reed proved that Q=0 is the threshold for the random graph to have a giant component. We prove that if $|Q|=O(n^{-1/3}{R}^{2/3})$ then, with high probability, the size of the largest component of the random graph will be of order $\Theta (n^{2/3}{R}^{-1/3})$. 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 $|Q|=O(n^{-1/3}{R}^{2/3})$.

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