  Different types of interactions coexist and coevolve to shape the structure
and function of a multiplex network. We propose here a general class of growth
models in which the various layers of a multiplex network coevolve through a
set of non-linear preferential attachment rules. We show, both numerically and
analytically, that by tuning the level of non-linearity these models allow to
reproduce either homogeneous or heterogeneous degree distributions, together
with positive or negative degree correlations across layers. In particular, we
derive the condition for the appearance of a condensed state in which one node
in each layer attracts an extensive fraction of all the edges.
