We investigate the phase transitions in a coupled system of Ising spins and a fluctuating network. Each spin interacts with $q$ neighbors through links of the rewiring network. The Ising spins and the network are in thermal contact with the heat baths at temperatures $T_S$ and $T_L$, respectively, so the whole system is driven out of equilibrium for $T_S\ne T_L$. The model is a generalization of the $q$-neighbor Ising model [A. Jędrzejewski et al., Phys. Rev. E 92, 052105 (2015)], which corresponds to the limiting case of $T_L=\infty$. Despite the mean-field nature of the interaction, the $q$-neighbor Ising model was shown to display a discontinuous phase transition for $q\ge 4$. Setting up the rate equations for the magnetization and the energy density, we obtain the phase diagram in the $T_S-T_L$ parameter space. The phase diagram consists of a ferromagnetic phase and a paramagnetic phase. The two phases are separated by a continuous phase transition belonging to the mean-field universality class or by a discontinuous phase transition with an intervening coexistence phase. The equilibrium system with $T_S=T_L$ falls into the former case while the $q$-neighbor Ising model falls into the latter case. At the tricritical point, the system exhibits the mean-field tricritical behavior. Our model demonstrates a possibility that a continuous phase transition turns into a discontinuous transition by a nonequilibrium driving. Heat flow induced by the temperature difference between two heat baths is also studied.

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• Entropy production
• Magnetic order
• Network phase transitions
• Phase diagrams
• Phase transitions
• Nonequilibrium lattice models