Convergence of Consensus Models With Stochastic Disturbances
We consider consensus algorithms in their most general setting and provide conditions under which such algorithms are guaranteed to converge, almost surely, to a consensus. Let {A(t), B(t)} ∈ R N×N be (possibly) stochastic, nonstationary matrices and {x(t), m(t)} 6 R N×1 be state and perturbation ve...
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Published in | IEEE transactions on information theory Vol. 56; no. 8; pp. 4101 - 4113 |
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Main Authors | , |
Format | Journal Article |
Language | English |
Published |
New York
IEEE
01.08.2010
The Institute of Electrical and Electronics Engineers, Inc. (IEEE) |
Subjects | |
Online Access | Get full text |
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Summary: | We consider consensus algorithms in their most general setting and provide conditions under which such algorithms are guaranteed to converge, almost surely, to a consensus. Let {A(t), B(t)} ∈ R N×N be (possibly) stochastic, nonstationary matrices and {x(t), m(t)} 6 R N×1 be state and perturbation vectors, respectively. For any consensus algorithm of the form x(t + 1) = A(t)x(t) + B(t)m(t), we provide conditions under which consensus is achieved almost surely, i.e., Pr-{lim t →∞ x(t) = c1} -1 for some c ∈ R. Moreover, we show that this general result subsumes recently reported results for specific consensus algorithms classes, including sum-preserving, nonsum-preserving, quantized, and noisy gossip algorithms. Also provided are the e-converging time for any such converging iterative algorithm, i.e., the earliest time at which the vector x(t) is ε close to consensus, and sufficient conditions for convergence in expectation to the average of the initial node measurements. Finally, mean square error bounds of any consensus algorithm of the form discussed above are presented. |
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Bibliography: | ObjectType-Article-2 SourceType-Scholarly Journals-1 ObjectType-Feature-1 content type line 23 |
ISSN: | 0018-9448 1557-9654 |
DOI: | 10.1109/TIT.2010.2050940 |