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We study a coevolving nonlinear voter model on a two-layer network.Coevolution stands for coupled dynamics of the state of the nodes and of thetopology of the network in each layer. The plasticity parameter p measures therelative time scale of the evolution of the states of the nodes and theevolution of the network by link rewiring. Nonlinearity of the interactions istaken into account through a parameter q that describes the nonlinear effect oflocal majorities, being q = 1 the marginal situation of the ordinary votermodel. Finally the connection between the two layers is measured by a degree ofmultiplexing `. In terms of these three parameters, p, q and ` we find a richphase diagram with different phases and transitions. When the two layers havethe same plasticity p, the fragmentation transition observed in a single layeris shifted to larger values of p plasticity, so that multiplexing avoidsfragmentation. Different plasticities for the two layers lead to new phasesthat do not exist in a coevolving nonlinear voter model in a single layer,namely an asymmetric fragmented phase for q > 1 and an active shattered phasefor q < 1. Coupling layers with dfferent types of nonlinearity, q1 < 1 and q2 >1, we can find two different transitions by increasing the plasticityparameter, a first absorbing transition with no fragmentation and a subsequentfragmentation transition.

Multilayer coevolution dynamics of the nonlinear voter model