We consider consensus algorithms in their most general setting and provide conditions under which such al- gorithms are guaranteed to converge, almost surely, to a con- sensus. Let {A(t) ,B (t) }∈ R N ×N be (possibly) random, non- stationary matrices and {x(t) ,m (t) }∈ 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) ,w e provide conditions under which consensus is achieved almost surely, i.e., Pr {limt→∞ 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, non-sum-preserving, quantized and noisy gossip algorithms. Also provided are the � -converging time for any such converging iterative algorithm, i.e., the earliest time at which the vector x(t) isclose to consensus, and sufficient conditions for convergence in expectation to the initial node measurements average.
['Tuncer C. Aysal', 'Kenneth E. Barner']
On the Convergence of Perturbed Non-Stationary Consensus Algorithms