Phase transitions in the q-voter model with noise on a duplex clique

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.