具有给定度序列的随机图的缩放窗口
The scaling window for a random graph with a given degree sequence
摘要：
考虑给定度序列${\cal D}$上满足一定条件的随机图.我们关注两个参数$Q=Q({\cal D}), R=R({\cal D})$。Molloy和Reed证明了Q=0是随机图具有巨分支的阈值。我们证明，如果$|Q|=O(n^{-1/3} R^{2/3})$，那么，以高概率，随机图的最大组件的大小将是顺序$Θ(n^{2/3}R^{-1/3})$。如果$|Q|$渐近大于$n^{-1/3}R^{2/3}$，则最大分量的大小渐近小于或大于$n^{2/3}R^{-1/3}$。因此，我们确定缩放窗口是$|Q|=O(n^{-1/3} R^{2/3})$。
英文摘要：
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 $Θ(n^{2/3}R^{-1/3})$. 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 $|Q|=O(n^{-1/3} R^{2/3})$.
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