  We define a dynamic model of random networks, where new vertices are
connected to old ones with a probability proportional to a sublinear function
of their degree. We first give a strong limit law for the empirical degree
distribution, and then have a closer look at the temporal evolution of the
degrees of individual vertices, which we describe in terms of large and
moderate deviation principles. Using these results, we expose an interesting
phase transition: in cases of strong preference of large degrees, eventually a
single vertex emerges forever as vertex of maximal degree, whereas in cases of
weak preference, the vertex of maximal degree is changing infinitely often.
Loosely speaking, the transition between the two phases occurs in the case when
a new edge is attached to an existing vertex with a probability proportional to
the root of its current degree.
