  We study a nonlinear q-voter model with stochastic noise, interpreted in the
social context as independence, on a duplex network. To study the role of the
multi-levelness we propose three methods of transferring the model from a mono-
to a multiplex network. They take into account two criteria -- one related to
the status of independence (LOCAL vs. GLOBAL) and one related to peer pressure
(AND vs. OR). In order to examine the influence of the presence of more than
one level in the social network, we perform simulations on a particularly
simple multiplex -- a duplex clique, which consists of two fully overlapped
complete graphs (cliques). Solving numerically the rate equation and
simultaneously conducting Monte Carlo simulations, we provide evidence that
even a simple rearrangement into a duplex topology may lead to significant
changes in the observed behavior. However, qualitative changes in the phase
transitions can be observed for only one of the considered rules -- LOCAL&AND.
For this rule the phase transition becomes discontinuous for q=5, whereas for a
monoplex such a behavior is observed for q=6. Interestingly, only this rule
admits construction of realistic variants of the model, in line with recent
social experiments.
