Synchronization in On-Off Stochastic Networks: Windows of Opportunity

We study dynamical networks whose topology and intrinsic parameters stochastically change, on a time scale that ranges from fast to slow. When switching is fast, the stochastic network synchronizes as long as synchronization in the averaged network, obtained by replacing the random variables by their mean, becomes stable. We apply a recently developed general theory of blinking systems to prove global stability of synchronization in the fast switching limit. We use a network of Lorenz systems to derive explicit probabilistic bounds on the switching frequency sufficient for the network to synchronize almost surely and globally. Going beyond fast switching, we consider networks of Rossler and Duffing oscillators and reveal unexpected windows of intermediate switching frequencies in which synchronization in the switching network becomes stable even though it is unstable in the averaged/fast-switching network.