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We analyze a modified kinetic Ising model, so called $q$-neighbor Isingmodel, with Metropolis dynamics, [Phys. Rev. E 92, 052105], on a duplex cliqueand a partially duplex clique. In the $q$-neighbor Ising model each spininteracts only with $q$ spins randomly chosen from its whole neighborhood. Inthe case of a duplex clique the change of a spin is allowed only if both levelssimultaneously induce this change. Due to the mean-field like nature of themodel we are able to derive the analytic form of transition probabilities andsolve the corresponding master equation. The existence of the second levelchanges dramatically the character of the phase transition. In the case of themonoplex clique, the $q$-neighbor Ising model exhibits continuous phasetransition for $q=3$, discontinuous phase transition for $q \ge 4$ and for$q=1$ and $q=2$ the phase transition is not observed. On the other hand, in thecase of the duplex clique continuous phase transitions are observed for allvalues of $q$, even for $q=1$ and $q=2$. Subsequently we introduce a partiallyduplex clique, parametrized by $r \in [0,1]$, which allows us to tune thenetwork from monoplex ($r=0$) to duplex ($r=1$). Such a generalized topology,in which a fraction $r$ of all nodes appear on both levels, allows to obtainthe critical value of $r=r^*(q)$ at which a tricriticality (switch fromcontinuous to discontinuous phase transition) appears.

Tricriticality in the $q$-neighbor Ising model on a partially duplex clique