Consensus value estimates in time-varying and directed networks

Abstract

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 cutbalance 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 timevanishing 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.

Consensus and influence power approximation in time‐varying and directed networks subject to perturbations

“…The present work is an extension of a preliminary version . Here, we add complementary results showing that the AIPV is smooth both with respect to time and the perturbation parameter ε , for any kind of smooth interaction weights.…”

“…Theorem is a substantial generalization of theorem 2 in the work of Martin et al to the case of general time‐varying communication weights depending on a parameter. A direct corollary of Theorem is the following.…”