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

TL;DR

This paper identifies the precise scaling window for the emergence of a giant component in a random graph with a given degree sequence, extending the understanding of phase transition behavior.

Contribution

It establishes the exact scaling window for the size of the largest component in such graphs, generalizing previous threshold results by Molloy and Reed.

Findings

The scaling window is characterized by |Q|=O(n^{-1/3} R^{2/3}).

Largest component size is Θ(n^{2/3} R^{-1/3}) within the window.

Outside the window, the component size deviates significantly from the critical scale.

Abstract

We consider a random graph on a given degree sequence ${\cal D}$, satisfying certain conditions. We focus on two parameters $Q=Q({\cal D}), R=R({\cal D})$. 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 $|Q|=O(n^{-1/3} R^{2/3})$.