Modulus consensus over time-varying digraphs

This paper considers a discrete-time modulus consensus model in which the interaction among a group of agents is described by a time-dependent, complex-valued, weighted digraph. It is shown that for any sequence of repeatedly jointly strongly connected digraphs, without any assumption on the structure of the complex-valued weights, the system asymptotically reaches modulus consensus. Sufficient conditions for exponential convergence to each possible type of limit states are provided. Specifically, it is shown that (1) if the sequence of complex-valued weighted digraphs is repeatedly jointly balanced with respect to the same type, the corresponding type of modulus consensus will be reached exponentially fast for almost all initial conditions; (2) if the sequence of complex-valued weighted digraphs is repeatedly jointly unbalanced, the system will converge to zero exponentially fast for all initial conditions.