AI 中文总结
研究具有给定度序列的多部随机图，证明其局部极限是多类型分支过程，去除前人假设，给出提取巨分支通用框架及多类型分支过程新生存准则。
AI 中文摘要
我们考虑具有给定度序列的多部随机图，包括不同划分内和不同划分间的情况。在一般假设下，我们证明该图的局部极限是一个多类型分支过程，确定只有当局部极限存活时巨分支才存在，并推断在超临界状态下典型距离在概率上是对数阶的。我们的分析去除了Gamarnik和Misra（2015）中的两个主要假设，首次考虑该模型的巨分支问题。特别是，我们不假设局部极限的不可约性，并提供了一个通用框架来提取巨分支，即使极限分支过程是可约的，我们希望这在其他情况下有用。我们还提供了一个新的更简单的多类型分支过程生存准则，当直接计算后代矩阵的谱半径可能困难时希望有用。
英文摘要
We consider multipartite random graphs with given degree sequences, within and across different partitions. Under general assumptions, we prove the local limit of this graph is a multi-type branching process, establish that a giant component exists only when the local limit survives, and deduce that the typical distance is of logarithmic order in probability in the supercritical regime. Our analysis removes two major assumptions from Gamarnik and Misra (2015), where the giant component problem for this model was first considered. In particular, we do not assume irreducibility of the local limit, and provide a general framework to extract giant components even when the limiting branching process is reducible, which we hope to be useful in other contexts. We also provide a new simpler survival criterion of multi-type branching processes, which we hope to be useful when direct calculation of the spectral radius of the offspring matrix may prove to be difficult.
Comments79 pages, comments welcome
