  We consider a general class of preferential attachment schemes evolving by a
reinforcement rule with respect to certain sublinear weights. In these schemes,
which grow a random network, the sequence of degree distributions is an object
of interest which sheds light on the evolving structures. In this article, we
use a fluid limit approach to prove a functional law of large numbers for the
degree structure in this class, starting from a variety of initial conditions.
The method appears robust and applies in particular to `non-tree' evolutions
where cycles may develop in the network. A main part of the argument is to
analyze an infinite system of coupled ODEs, corresponding to a rate formulation
of the law of large numbers limit, in terms of $C_0$-semigroup/dynamical
systems methods. These results also resolve a question in Chung, Handjani and
Jungreis (2003).
