Consensus value estimates in time-varying and directed networks

The paper provides consensus value estimates in multi-agent systems with directed and time-varying interaction networks. First, we prove general results regarding the asymptotic consensus value which is obtained as a convex combination of the initial states. It is shown that the cut-balance assumption guarantees a strictly positive lower bound on the convex combination components. This means that each agent plays a non vanishing role in the asymptotic consensus value. Second, we analyze the case where interaction weights vary uniformly over time. Finally, we study the effect of time-vanishing perturbations on the systems with uniform variation of the interaction weights. We show that the convex combination components vary smoothly with the perturbation under smooth and sufficiently fast vanishing perturbations. Moreover, we show that in this case, these components reach a limit when time goes to infinity. We also provide an example where this limit does not exist when the perturbation does not respect the fast vanishing assumption although the system itself converges to a consensus. Some numerical examples illustrate our results.