  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.
