  The Ising model on networks plays a fundamental role as a testing ground for
understanding cooperative phenomena in complex systems. Here we solve the
synchronous dynamics of the Ising model on random graphs with an arbitrary
degree distribution in the high-connectivity limit. Depending on the
distribution of the threshold noise that governs the microscopic dynamics, the
model evolves to nonequilibrium stationary states. We obtain an exact dynamical
equation for the distribution of local magnetizations, from which we find the
critical line that separates the paramagnetic from the ferromagnetic phase. For
random graphs with a negative binomial degree distribution, we demonstrate that
the stationary critical behavior as well as the long-time critical dynamics of
the first two moments of the local magnetizations depend on the distribution of
the threshold noise. In particular, for an algebraic threshold noise, these
critical properties are determined by the power-law tails of the distribution
of thresholds. We further show that the relaxation time of the average
magnetization inside each phase exhibits the standard mean-field critical
scaling. The values of all critical exponents considered here are independent
of the variance of the negative binomial degree distribution. Our work
highlights the importance of certain details of the microscopic dynamics for
the critical behavior of nonequilibrium spin systems.
