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We study synchronization in scalar nonlinear systems connected over a linearnetwork with stochastic uncertainty in their interactions. We provide asufficient condition for the synchronization of such network systems expressedin terms of the parameters of the nonlinear scalar dynamics, the second andlargest eigenvalues of the mean interconnection Laplacian, and the variance ofthe stochastic uncertainty. The sufficient condition is independent of networksize thereby making it attractive for verification of synchronization in alarge size network. The main contribution of this paper is to provideanalytical characterization for the interplay of roles played by the internaldynamics of the nonlinear system, network topology, and uncertainty statisticsin network synchronization. We show there exist important tradeoffs betweenthese various network parameters necessary to achieve synchronization. We showfor nearest neighbor networks with stochastic uncertainty in interactions thereexists an optimal number of neighbors with maximum margin for synchronization.This proves in the presence of interaction uncertainty, too many connectionsamong network components is just as harmful for synchronization as the lack ofconnection. We provide an analytical formula for the optimal gain required toachieve maximum synchronization margin thereby allowing us to compare variouscomplex network topology for their synchronization property.

Limitations and tradeoffs in synchronization of large-scale networks with uncertain links