Convergence of Consensus Models With Stochastic Disturbances

• Published in: IEEE transactions on information theory Vol. 56; no. 8; pp. 4101 - 4113
• Main Authors: Aysal, Tuncer Can, Barner, Kenneth E
• Format: Journal Article
• Language: English
• Published: New York IEEE 01.08.2010; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Algorithms; Application software; Computational modeling; Convergence; Convergence in expectation; convergence of random sequences; convergence to consensus; distributed average consensus; Disturbances; gossip algorithms; Information processing; Information systems; Iterative algorithms; Mathematical analysis; Matrix; mean square error; Mean square error methods; noisy gossip algorithms; nonsum-preserving gossip algorithms; Perturbation methods; quantized gossip algorithms; Random sequences; Signal processing algorithms; State vectors; Stochastic models; Stochastic processes; Stochasticity; Sufficient conditions; sum-preserving gossip algorithms; Time measurement; Vectors (mathematics)
• ISSN: 0018-9448; 1557-9654
• DOI: 10.1109/TIT.2010.2050940

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.