This talk discusses the influence of network structure on the dynamics of networks with fast on-off stochastic connections. It is investigated to what extent the trajectories of a stochastically switched (blinking) network follow the corresponding trajectories of its averaged analog where the stochastic connection are replaced with static ones. Four cases have to be distinguished, depending on whether or not the averaged system has a unique attractor and whether or not the attractors are invariant under the dynamics of the blinking system. The corresponding asymptotic behavior of the trajectories of the blinking system is described and illustrative examples are given.

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