  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 that the
whole system is driven out of equilibrium for $T_S \neq T_L$. The model is a
generalization of the $q$-neighbor Ising model, 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 $q\geq 4$. 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.
