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Description
The $q$-neighbor Ising model is considered on multiplex networks with two layers in the form of identical random graphs, in which only a fraction of nodes belongs to both layers, forming the overlap. In this model the probability of the spin flip for a node belonging only to one layer is given by the Metropolis-like formula with the local field depending on the states of its $q$ randomly chosen neighbors within this layer, and for a node belonging to the overlap is a product of the above-mentioned probabilities for the two layers. Critical properties of the model for varying temperature depend on the size of the neighborhood $q$ and of the overlap $r$ as well as on the mean degree of nodes $ \langle k \rangle$ within the layers. For large $\langle k \rangle$ results for the model on multiplex networks with layers in the form of complete graphs are reproduced which can be explained using mean-field approximation [A. Chmiel et al., Phys. Rev. E 96, 062137 (2017)]. In particular, for $q=2$ for large enough $r$ including complete overlap first-order ferromagnetic transition occurs, for a narrow interval of smaller $r$ the paramagnetic and ferromagnetic phases coexist for temperature decreasing to zero, and for small $r$ the paramagnetic phase is stable for any temperature; and for $q \ge 4$ the ferromagnetic transition is first-order for small $r$ and second-order for larger $r$ including complete overlap. As $\langle k \rangle$ is decreased, for $q=2$ first-order transition and coexistence of the paramagnetic and ferromagnetic phase occurs at smaller $r$, and for large and complete overlap second-order ferromagnetic transition appears; and for $q\ge 4$ the interval of small $r$ for which first-order transition occurs becomes narrower. Results of Monte Carlo simulations for the model on multiplex networks with layers in the form of random regular graphs show good quantitative agreement with predictions of the homogeneous pair approximation. Such agreement is remarkable since the latter approximation takes into account only heterogeneity of the nodes due to their location within or outside the overlap, while the density of active links connecting spins with opposite orientation is assumed uniform within each layer.
