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The $q$-neighbor Ising model for the opinion formation on multiplex networkswith two layers in the form of random graphs (duplex networks), the partialoverlap of nodes, and LOCAL\&AND spin update rule was investigated by means ofthe pair approximation and approximate Master equations as well as Monte Carlosimulations. Both analytic and numerical results show that for different fixedsizes of the $q$-neighborhood and finite mean degrees of nodes within thelayers the model exhibits qualitatively similar critical behavior as theanalogous model on multiplex networks with layers in the form of completegraphs. However, as the mean degree of nodes is decreased the discontinuousferromagnetic transition, the tricritical point separating it from thecontinuous transition and the possible coexistence of the paramagnetic andferromagnetic phases at zero temperature occur for smaller relative sizes ofthe overlap. Predictions of the simple homogeneous pair approximationconcerning the critical behavior of the model under study show good qualitativeagreement with numerical results; predictions based on the approximate Masterequations are usually quantitatively more accurate, but yet not exact. Twoversions of the heterogeneous pair approximation are also derived for the modelunder study, which, surprisingly, yield predictions only marginally differentor even identical to those of the simple homogeneous pair approximation. Ingeneral, predictions of all approximations show better agreement with theresults of Monte Carlo simulations in the case of continuous than discontinuousferromagnetic transition.

The $q$-neighbor Ising model on multiplex networks with partial overlap of nodes