K核问题的简单解决方案
A simple solution to the k-core problem
摘要：
研究了具有给定度序列的n阶随机（多）图的k-核。我们让n趋于无穷大。然后，在度序列的一些正则性条件下，我们给出了度序列的渐近形状的条件，这意味着有很高的概率k-核是空的，以及其他条件，这意味着有很高的概率k-核是非空的，其顶点和边集的大小满足大数定律;在适当的假设下，这是仅有的两种可能性。特别地，我们恢复了Pittel，Spencer和Wormald关于G（n，p）和G（n，m）中k-核的存在性和大小的结果.我们的方法是基于独立随机变量的经验分布的性质，并导致简单的证明。
英文摘要：
We study the k-core of a random (multi)graph on n vertices with a given degree sequence. We let n tend to infinity. Then, under some regularity conditions on the degree sequences, we give conditions on the asymptotic shape of the degree sequence that imply that with high probability the k-core is empty, and other conditions that imply that with high probability the k-core is non-empty and the sizes of its vertex and edge sets satisfy a law of large numbers; under suitable assumptions these are the only two possibilities. In particular, we recover the result by Pittel, Spencer and Wormald on the existence and size of a k-core in G(n,p) and G(n,m). Our method is based on the properties of empirical distributions of independent random variables, and leads to simple proofs.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
