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We analyze a nonlinear $q$-voter model with stochastic noise, interpreted inthe social context as independence, on a duplex network. The size of the lobby$q$ (i.e., the pressure group) is a crucial parameter that changes the behaviorof the system. The $q$-voter model has been applied on multiplex networks in aprevious work [Phys. Rev E. 92. 052812. (2015)], and it has been shown that thecharacter of the phase transition depends on the number of levels in themultiplex network as well as the value of $q$. Here we study phase transitioncharacter in the case when on each level of the network the lobby size isdifferent, resulting in two parameters $q_1$ and $q_2$. We find evidence ofsuccessive phase transitions when a continuous phase transition is followed bya discontinuous one or two consecutive discontinuous phases appear, dependingon the parameter. When analyzing this system, we even encounter mixed-order (orhybrid) phase transition. We perform simulations and obtain supportinganalytical solutions on a simple multiplex case - a duplex clique, whichconsists of two fully overlapped complete graphs (cliques).

A veritable zoology of successive phase transitions in the asymmetric $q$-voter model on multiplex networks