Consensus in networks under transmission delays and the normalized Laplacian

We study the consensus problem on directed and weighted networks in the presence of 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 to the past states of its neighbors. The connection structure of the network is described by a normalized Laplacian matrix. We show that consensus is achieved if and only if the underlying graph contains a spanning tree. Furthermore, this statement holds independently of the value of the delay, in contrast to the case of processing delays. We also calculate the consensus value and show that, unlike the case of processing delays, the consensus value is determined not just by the initial states of the nodes at time zero, but also on their past history over an interval of time.

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