Stochastic Pinning Controllability of Noisy Complex Networks

Our ability to coordinate the behavior in networks of complex dynamical systems is often challenged by the presence of noise affecting the individual dynamics and the communication links. In the literature, conservative global conditions guaranteeing the almost sure convergence toward the desired trajectory of a virtual node, the pinner, have been derived. In this article, we identify the minimal conditions on the individual dynamics, interconnection topology, and noise intensities, so that the network exponentially converges onto the pinner's trajectory. Specifically, we broaden the master stability function approach to deal with networks of coupled stochastic differential equations, and provide necessary and sufficient conditions for local exponential pinning controllability of networks of stochastic systems. Interestingly, our analyses show that noise can be either beneficial or detrimental for pinning controllability, depending on how it diffuses in each node. Our analytical findings are illustrated with representative numerical examples.