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We investigate a kinetic Ising model with the Metropolis algorithm on the temporal directed random $q$-regular graph. We introduce a rewiring time $\tau$ as a parameter of the model and show that there is a critical $\tau=\tau^*$ above which the model exhibits continuous order-disorder phase transition and below which the transition is discontinuous. We discuss our findings in a light of the very recent results obtained for a generalized kinetic Ising model on quenched and annealed networks.

Phase Transitions in the Kinetic Ising Model on the Temporal Directed Random Regular Graph