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In this paper, we study the problem of synchronization with stochasticinteraction among network components. The network components dynamics isnonlinear and modeled in Lure form with linear stochastic interaction amongnetwork components. To study this problem we first prove the stochastic versionof Positive Real Lemma (PRL). The stochastic PRL result is then used to providesufficient condition for synchronization of stochastic network system. Thesufficiency condition for synchronization, is a function of nominal (mean)coupling Laplacian eigenvalues and the statistics of link uncertainty in theform of coefficient of dispersion (CoD). Contrary to the existing literature onnetwork synchronization, our results indicate that both the largest and thesecond smallest eigenvalue of the mean Laplacian play an important role insynchronization of stochastic networks. Robust control-based small-gaininterpretation is provided for the derived sufficiency condition which allow usto define the margin of synchronization. The margin of synchronization is usedto understand the important tradeoff between the component dynamics, networktopology, and uncertainty characteristics. For a special class of networksystem connected over torus topology we provide an analytical expression forthe tradeoff between the number of neighbors and the dimension of the torus.Similarly, by exploiting the identical nature of component dynamicscomputationally efficient sufficient condition independent of network size isprovided for general class of network system. Simulation results for network ofcoupled oscillators with stochastic link uncertainty are presented to verifythe developed theoretical framework.

Mean sqaure synchronization in large scale nonlinear networks with uncertain links