A Necessary and Sufficient Condition for Consensus Over Random Networks
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graph theory
linear systems
stochastic systems
time-varying systems
average weight matrix
random graph process
random networks
stochastic discrete-time linear dynamical systems
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Abstract
We consider the consensus problem for stochastic discrete-time linear dynamical systems. The underlying graph of such systems at a given time instance is derived from a random graph process, independent of other time instances. For such a framework, we present a necessary and sufficient condition for almost sure asymptotic consensus using simple ergodicity and probabilistic arguments. This easily verifiable condition uses the spectrum of the average weight matrix. Finally, we investigate a special case for which the linear dynamical system converges to a fixed vector with probability 1.
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Copyright 2008 IEEE. Reprinted from: Tahbaz-Salehi, A.; Jadbabaie, A., "A Necessary and Sufficient Condition for Consensus Over Random Networks," Automatic Control, IEEE Transactions on , vol.53, no.3, pp.791-795, April 2008 URL: http://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=4484213&isnumber=4484183 This material is posted here with permission of the IEEE. Such permission of the IEEE does not in any way imply IEEE endorsement of any of the University of Pennsylvania's products or services. Internal or personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution must be obtained from the IEEE by writing to pubs-permissions@ieee.org. By choosing to view this document, you agree to all provisions of the copyright laws protecting it.