The consensus problem in networks with transmission delays
Fatihcan M. Atay
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Submission date: 18. Jan. 2013 (revised version: August 2013)
published in: Philosophical transactions of the Royal Society / A, 371 (2013) 1999, art-no. 20120460
DOI number (of the published article): 10.1098/rsta.2012.0460
Keywords and phrases: Consensus, distributed delay, graph, normalized Laplacian, synchronization
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 to the past states of its neighbors. 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 past 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.