We consider models of evolving networks \(\left\{ {\mathcal {G}}_n:n\ge 0\right\} \) modulated by two parameters: an attachment function \(f:{\mathbb {N}}_0 \rightarrow {\mathbb {R}}_+\) and a (possibly random) attachment sequence \(\left\{ m_i:i\ge 1\right\} \). Starting with a single vertex, at each discrete step \(i\ge 1\) a new vertex \(v_i\) enters the system with \(m_i\ge 1\) edges which it sequentially connects to a pre-existing vertex \(v\in {\mathcal {G}}_{i-1}\) with probability proportional to \(f(\text{ degree }(v))\). We consider the problem of emergence of persistent hubs: existence of a finite (a.s.) time \(n^*\) such that for all \(n\ge n^*\) the identity of the maximal degree vertex (or in general the K largest degree vertices for \(K\ge 1\)) does not change. We obtain general conditions on f and \(\left\{ m_i:i\ge 1\right\} \) 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 deg ree 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 form the technical foundations for the main results.

我们考虑由两个参数调制的演化网络模型\(\left\{ {\mathcal {G}}_n:n\ge 0\right\} \)：一个附件函数\(f:{\mathbb {N}} _0 \rightarrow {\mathbb {R}}_+\)和（可能是随机的）附件序列\(\left\{ m_i:i\ge 1\right\} \)。从单个顶点开始，在每个离散步骤\(i\ge 1\)一个新顶点\(v_i\)进入系统，其边依次连接到预先存在的顶点\ (m_i\ge 1\) (v\in {\mathcal {G}}_{i-1}\)概率与\(f(\text{ degree }(v))\)成正比。我们考虑持久集线器的出现问题：有限（作为）时间的存在\(n^*\)使得对于所有\(n\ge n^*\)最大度顶点的身份（或一般来说\(K\ge 1\)的K 个最大度顶点）不会改变. 我们在f和\(\left\{ m_i:i\ge 1\right\} \)上获得了持久集线器出现以及持久集线器未能出现的一般条件。在缺乏持久性的情况下，对于树的特定情况（\(m_i\equiv 1\)对于所有i)，我们推导出最大度数和最大度数顶点的索引（当前度数最大的顶点进入系统的时间）的渐近线，以了解最大度数顶点随着网络的发展而移动。分析中的关键作用是由离散时间模型的连续时间嵌入构建的反速率加权鞅。这种鞅的渐近线，包括浓度不等式和适度的偏差，构成了主要结果的技术基础。