The consensus problem in networks with transmission delays

We study discrete- and continuous-time consensus problems on networks in the presence of distributed time delays. We focus on information transmission delays, as opposed to information processing delays, so that each node of the network compares its current state with the past states of its neighbours. We consider directed and weighted networks where the connection structure is described by a normalized Laplacian matrix and show that consensus is achieved if and only if the underlying graph contains a directed spanning tree. This statement holds independently of the transmission delays, which is in contrast to the case of processing delays. Furthermore, we calculate the consensus value explicitly, and show that it is determined by the history of the system over an interval of time, unlike the case of processing delays where the consensus value depends only on the initial state of the system at time zero. This provides the consensus algorithm with improved robustness against noise.