论文标题
枢纽在增长的随机网络中的持久性
Persistence of hubs in growing random networks
论文作者
论文摘要
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 $\{m_i:i\geq 1\}$.从单个顶点开始，在每个离散步骤$ i \ geq 1 $ a新顶点$ v_i $都以$ m_i \ geq 1 $ edges进入系统，它将其顺序连接到预先存在的顶点$ v \ in \ Mathcal {g} {g} _ {g} _ {g} _ {i-1} $ cobitability prabipitionalitionaltion $ f（$ f（$ f）$ f（$ f（$））我们考虑了持续枢纽的出现问题：存在有限的（A.S.）时间$ n^*$，以使所有$ n \ geq n^*$对于最大度顶点的身份（或一般而言，$ k $ $ k $最大程度的顶点的$ k \ geq 1 $）都不会改变。我们在$ f $和$ \ {m_i：i \ geq 1 \} $上获得一般条件，在该$下出现了持续的集线器，并且在其下，持续集线器未能出现的那些。在缺乏持久性的情况下，对于树木的具体情况（所有$ i $的$ M_i \ equiv 1 $），我们在最大程度的最大程度和最大程度顶点的索引（当前具有最大最大程度的最大程度输入系统输入最大程度的顶点的时间）以了解最大程度的最大程度vertex as vertex as the Maximal vertex as the Maximal vertex。分析中的关键作用是通过离散时间模型的连续嵌入时间构建的反率加权martingale发挥的。该群岛的渐近学，包括集中不平等和中等偏差，在模型分析中起着重要作用。
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.
