增长随机网络中枢纽的持久性
Persistence of hubs in growing random networks
摘要：
我们考虑模型的演变网络$\{\mathcal{G}_n:n\geq 0\}$调制的两个参数：附件功能$f:\mathbb{N}_0\to\mathbb{R}_+$和（可能是随机的）附件序列$\{m_i:i\geq 1\}$。从单个顶点开始，在每个离散步骤$i\geq 1$，新顶点$v_i$进入具有$m_i\geq 1$边的系统，其以与$f(\operatorname{degree}(v))$成比例的概率顺序地连接到预先存在的顶点$v\in \mathcal{G}_{i-1}$。我们考虑持久枢纽的出现问题：存在一个有限的（a.s.）时间$n^*$，使得对于所有$n\geq n^*$，最大度顶点（或一般地，对于$K\geq 1$，$K$最大度顶点）的身份不改变。我们得到了$f$和$\{m_i:i\geq 1\}$上持久中枢出现的一般条件，以及持久中枢不出现的一般条件。在缺乏持久性的情况下，对于树的特定情况（$m_i\equiv 1$为所有$i$），我们推导出最大度和最大度顶点的指数（当前最大度的顶点进入系统的时间）的渐近性，以了解最大度顶点随着网络演化的运动。在分析中的一个关键作用是由一个连续时间嵌入的离散时间模型构造的逆率加权鞅。这个鞅的渐近性，包括集中不等式和中等偏差，在模型的分析中起着重要的作用。
英文摘要：
We consider models of evolving networks $\{\mathcal{G}_n:n\geq 0\}$ modulated by two parameters: an attachment function $f:\mathbb{N}_0\to\mathbb{R}_+$ and a (possibly random) attachment sequence $\{m_i:i\geq 1\}$. Starting with a single vertex, at each discrete step $i\geq 1$ a new vertex $v_i$ enters the system with $m_i\geq 1$ edges which it sequentially connects to a pre-existing vertex $v\in \mathcal{G}_{i-1}$ with probability proportional to $f(\operatorname{degree}(v))$. We consider the problem of emergence of persistent hubs: existence of a finite (a.s.) time $n^*$ such that for all $n\geq n^*$ the identity of the maximal degree vertex (or in general the $K$ largest degree vertices for $K\geq 1$) does not change. We obtain general conditions on $f$ and $\{m_i:i\geq 1\}$ under which a persistent hub emerges, and also those under which a persistent hub fails to emerge. In the case of lack of persistence, for the specific case of trees ($m_i\equiv 1$ for all $i$), we derive asymptotics for the maximal degree and the index of the maximal degree vertex (time at which the vertex with current maximal degree entered the system) to understand the movement of the maximal degree vertex as the network evolves. A key role in the analysis is played by an inverse rate weighted martingale constructed from a continuous time embedding of the discrete time model. Asymptotics for this martingale, including concentration inequalities and moderate deviations, play a major role in the analysis of the model.
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