一些随机图的邻近性与远程邻近性
Contiguity and remote contiguity of some random graphs
AI总结：
本文利用远程邻近性概念证明在受更强烈扰动的Erdős-Rényi随机图中连通性等渐近性质得以保持，并指出该技术可推广至大图极限等一般渐近性质分析。
摘要：
随机图序列的渐近性质，如Erdős-Rényi图中巨连通分量的出现或完全连通性，通常是在定义参数的非常特定的选择下推导的。由此产生的问题是：在多大程度上可以扰动这些参数选择而不失去渐近性质。将$(P_n)$和$(Q_n)$记为两个图分布序列，渐近等价性（全变差收敛）和邻近性（$P_n(A_n)=o(1) \implies Q_n(A_n)=o(1)$）已被(Janson, 2010)等人考虑；这里我们使用所谓的远程邻近性（对于某个固定的$a_n\downarrow 0$，$P_n(A_n)=o(a_n) \implies Q_n(A_n)=o(1)$）来证明在受更强烈扰动的Erdős-Rényi图中连通性得以保持。我们在此用随机图展示的技术，可推广至一般的渐近性质，例如更复杂的大图极限、标度极限、大样本极限等。
英文摘要：
Asymptotic properties of random graph sequences, like occurrence of a giant component or full connectivity in Erdős-Rényi graphs, are usually derived with very specific choices for defining parameters. The question arises to which extent those parameters choices may be perturbed, without losing the asymptotic property. Writing $(P_n)$ and $(Q_n)$ for two sequences of graph distributions, asymptotic equivalence (convergence in total-variation) and contiguity ($P_n(A_n)=o(1) \implies Q_n(A_n)=o(1)$) have been considered by (Janson, 2010) and others; here we use so-called remote contiguity (for some fixed $a_n\downarrow 0$, $P_n(A_n)=o(a_n) \implies Q_n(A_n)=o(1)$) to show that connectivity properties are preserved in more heavily perturbed Erdős-Rényi graphs. The techniques we demonstrate with random graphs here, extend to general asymptotic properties, e.g. in more complex large-graph limits, scaling limits, large-sample limits, etc.
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。
