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

A scaling limit for the degree distribution in sublinear preferential attachment schemes