具有无穷平均度的随机图中的距离
Distances in random graphs with infinite mean degrees
摘要：
我们研究具有独立同分布的随机图。度序列，其分布函数$F$的尾部随指数$τ\in (1,2)$规则地变化。因此，度具有无限的意义。这样的随机图可以作为复杂网络的模型，其中度幂律被观察到。研究了当图中有$N$节点时，任意两个节点之间的最小边数，也称为图距离或跳数.本文是三篇论文的续集的一部分。另外两篇论文分别研究了$τ\in (2,3)$和$τ\in (3,\infty),$的情况。本文的主要结果是图距离收敛$τ\in (1,2)$到一个极限随机变量的概率质量只在点2和3。我们还考虑的情况下，我们的条件下，最多$N^α$的一些$α>0.$对于$τ^{-1}<α<(τ-1)^{-1}$，跳数收敛到3的概率，而对于$α>(τ-1)^{-1}$，跳数收敛到相同的限制，为无条件的程度。我们的结果给出了令人信服的渐近跳数时，平均度是无限的，使用极值理论。
英文摘要：
We study random graphs with an i.i.d. degree sequence of which the tail of the distribution function $F$ is regularly varying with exponent $τ\in (1,2)$. Thus, the degrees have infinite mean. Such random graphs can serve as models for complex networks where degree power laws are observed. The minimal number of edges between two arbitrary nodes, also called the graph distance or the hopcount, in a graph with $N$ nodes is investigated when $N\to \infty$. The paper is part of a sequel of three papers. The other two papers study the case where $τ\in (2,3)$, and $τ\in (3,\infty),$ respectively. The main result of this paper is that the graph distance converges for $τ\in (1,2)$ to a limit random variable with probability mass exclusively on the points 2 and 3. We also consider the case where we condition the degrees to be at most $N^α$ for some $α>0.$ For $τ^{-1}<α<(τ-1)^{-1}$, the hopcount converges to 3 in probability, while for $α>(τ-1)^{-1}$, the hopcount converges to the same limit as for the unconditioned degrees. Our results give convincing asymptotics for the hopcount when the mean degree is infinite, using extreme value theory.
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