The component sizes of a critical random graph with given degree sequence
The component sizes of a critical random graph with given degree sequence
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给定度数序列的临界随机图的分量大小
DOI：
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发表时间：
2010
期刊：
影响因子：
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通讯作者：
Adrien Joseph
中科院分区：
文献类型：
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作者：
Adrien Joseph
考虑由配置模型构造的具有 $n$ 个顶点的临界随机多重图 $mathcal{G}_n$ ，使其顶点度数是具有相同分布的独立随机变量 $
u$（临界性是指$的二阶矩
u$ 是有限的并且等于其一阶矩的两倍）。我们指定 $mathcal{G}_n$ 组件大小的有序序列的缩放限制，因为 $n$ 在不同情况下趋于无穷大。当$
u$ 具有有限的三阶矩，按 $n^{-2/3}$ 重新缩放的分量大小收敛到抛物线漂移高于过去最小值的布朗运动的偏移长度，而当 $
u$ 是指数为 $gammain(3,4)$ 的幂律分布，按 $n^{-(gamma -2)/(gamma-1)}$ 重新调整的分量大小收敛到某个非平凡漂移过程的偏移长度，其独立增量高于过去的最小值。当 $ 时，我们推导出临界随机简单图的分量大小的渐近行为
u$ 具有有限的第三矩。
Consider a critical random multigraph $mathcal{G}_n$ with $n$ vertices constructed by the configuration model such that its vertex degrees are independent random variables with the same distribution $
u$ (criticality means that the second moment of $
u$ is finite and equals twice its first moment). We specify the scaling limits of the ordered sequence of component sizes of $mathcal{G}_n$ as $n$ tends to infinity in different cases. When $
u$ has finite third moment, the components sizes rescaled by $n^{-2/3}$ converge to the excursion lengths of a Brownian motion with parabolic drift above past minima, whereas when $
u$ is a power law distribution with exponent $gammain(3,4)$, the components sizes rescaled by $n^{-(gamma -2)/(gamma-1)}$ converge to the excursion lengths of a certain nontrivial drifted process with independent increments above past minima. We deduce the asymptotic behavior of the component sizes of a critical random simple graph when $
u$ has finite third moment.