  We investigate a kinetic Ising model with several single-spin flip dynamics
(including Metropolis and heat-bath) on quenched and annealed random regular
graphs. As expected, on the quenched structures all proposed algorithms
reproduce the same results since the conditions for the detailed balance and
the Boltzmann distribution in an equilibrium are satisfied. However, on the
annealed graphs situation is far less clear -- the network annealing disturbs
the equilibrium moving the system away from it. Consequently, distinct dynamics
lead to different steady states. We show that some algorithms are more
resistant to the annealed disorder, which causes only small quantitative
changes in the model behavior. On the other hand, there are dynamics for which
the influence of annealing on the system is significant, and qualitative
changes arise like switching the type of phase transition from continuous to
discontinuous one. We try to identify features of the proposed dynamics which
are responsible for the above phenomenon.
