Network consensus under distributed transmission delays

We study the linear consensus problem in weighted and directed networks under signal transmission delays. In contrast to signal processing delays, the average of the initial conditions is not an invariant of the dynamics. Consequently, the consensus value depends on the delay distribution. On the other hand, the condition for convergence to consensus does not depend on delays but requires the existence of a directed spanning tree in the network. We consider both discrete and distributed delays and for each case calculate the consensus value explicitly.