  We study synchronization in scalar nonlinear systems connected over a linear
network with stochastic uncertainty in their interactions. We provide a
sufficient condition for the synchronization of such network systems expressed
in terms of the parameters of the nonlinear scalar dynamics, the second and
largest eigenvalues of the mean interconnection Laplacian, and the variance of
the stochastic uncertainty. The sufficient condition is independent of network
size thereby making it attractive for verification of synchronization in a
large size network. The main contribution of this paper is to provide
analytical characterization for the interplay of roles played by the internal
dynamics of the nonlinear system, network topology, and uncertainty statistics
in network synchronization. We show there exist important tradeoffs between
these various network parameters necessary to achieve synchronization. We show
for nearest neighbor networks with stochastic uncertainty in interactions there
exists an optimal number of neighbors with maximum margin for synchronization.
This proves in the presence of interaction uncertainty, too many connections
among network components is just as harmful for synchronization as the lack of
connection. We provide an analytical formula for the optimal gain required to
achieve maximum synchronization margin thereby allowing us to compare various
complex network topology for their synchronization property.
