  We characterize synchronization phenomenon in discrete-time, discrete-state
random dynamical systems, with random and probabilistic Boolean networks as
particular examples. In terms of multiplicative ergodic properties of the
induced linear cocycle, we show such a random dynamical system with finite
state synchronizes if and only if the Lyapunov exponent $0$ has simple
multiplicity. For the case of countable state space, characterization of
synchronization is provided in terms of the spectral subspace corresponding to
the Lyapunov exponent $-\infty$. In addition, for both cases of finite and
countable state spaces, the mechanism of partial synchronization is described
by partitioning the state set into synchronized subsets. Applications to
biological networks are also discussed.
