  In this paper, we present new results for the synchronization and consensus
of networks described by Ito stochastic differential equations. From the
methodological viewpoint, our results are based on the use of stochastic
Lyapunov functions. This approach allowed us to consider networks where nodes
dynamics can be nonlinear and non-autonomous and where noise is not just
additive but rather its diffusion can be nonlinear and depend on the network
state. We first present a sufficient condition on the coupling strength and
topology ensuring that a network synchronizes (fulfills consensus) despite
noise. Then, we show that noise can be useful, and present a result showing how
to design noise so that it induces synchronization/consensus. Motivated by our
current research in Smart Cities and Internet of Things, we also illustrate the
effectiveness of our approach by showing how our results can be used to
analyze/control the onset of synchronization in noisy networks and to study
collective decision processes.
