  In this paper, we study the problem of synchronization with stochastic
interaction among network components. The network components dynamics is
nonlinear and modeled in Lure form with linear stochastic interaction among
network components. To study this problem we first prove the stochastic version
of Positive Real Lemma (PRL). The stochastic PRL result is then used to provide
sufficient condition for synchronization of stochastic network system. The
sufficiency condition for synchronization, is a function of nominal (mean)
coupling Laplacian eigenvalues and the statistics of link uncertainty in the
form of coefficient of dispersion (CoD). Contrary to the existing literature on
network synchronization, our results indicate that both the largest and the
second smallest eigenvalue of the mean Laplacian play an important role in
synchronization of stochastic networks. Robust control-based small-gain
interpretation is provided for the derived sufficiency condition which allow us
to define the margin of synchronization. The margin of synchronization is used
to understand the important tradeoff between the component dynamics, network
topology, and uncertainty characteristics. For a special class of network
system connected over torus topology we provide an analytical expression for
the tradeoff between the number of neighbors and the dimension of the torus.
Similarly, by exploiting the identical nature of component dynamics
computationally efficient sufficient condition independent of network size is
provided for general class of network system. Simulation results for network of
coupled oscillators with stochastic link uncertainty are presented to verify
the developed theoretical framework.
