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

We analyze a modified kinetic Ising model, so called $q$-neighbor Ising model, with Metropolis dynamics, [Phys. Rev. E 92, 052105], on a duplex clique and a partially duplex clique. In the $q$-neighbor Ising model each spin interacts only with $q$ spins randomly chosen from its whole neighborhood. In the case of a duplex clique the change of a spin is allowed only if both levels simultaneously induce this change. Due to the mean-field like nature of the model we are able to derive the analytic form of transition probabilities and solve the corresponding master equation. The existence of the second level changes dramatically the character of the phase transition. In the case of the monoplex clique, the $q$-neighbor Ising model exhibits continuous phase transition 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 the case of the duplex clique continuous phase transitions are observed for all values of $q$, even for $q=1$ and $q=2$. Subsequently we introduce a partially duplex clique, parametrized by $r \in [0,1]$, which allows us to tune the network 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 obtain the critical value of $r=r^*(q)$ at which a tricriticality (switch from continuous to discontinuous phase transition) appears.