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Convergence time for unbiased quantized consensus
 Proc. 52nd IEEE Conference on Decision and Control. CDC’13, Dec 1013, 2013
"... We revisit the quantized consensus problem on undirected connected graphs, and obtain some strong results on expected time to convergence. This is unbiased consensus, because the edges emanating from a node have equal probability of being selected. The paper first develops an approach that bounds th ..."
Abstract

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We revisit the quantized consensus problem on undirected connected graphs, and obtain some strong results on expected time to convergence. This is unbiased consensus, because the edges emanating from a node have equal probability of being selected. The paper first develops an approach that bounds the expected convergence time of the underlying discretetime dynamics. The bounds are tight for some simple networks when there exists some symmetry in the network. Following this, the paper provides a tight expression for the expected convergence time of unbiased quantized consensus over general but fixed networks. We show that the expected convergence time can be expressed in terms of the effective resistances of the associated Cartesian product graph. The approach adopted in the paper uses the theory of harmonic functions for reversible Markov chains. Finally, we extend our results to bound the expected convergence time of the underlying dynamics in timevarying networks. In spite of lack of reversibility for timevarying networks, we prove a tight upper bound for expected convergence time of the dynamics using the spectral representation of the networks. Index Terms
Fastest Expected Time to Mixing for a Markov Chain on a Directed Graph
"... For an irreducible stochastic matrix T, the Kemeny constant K(T) measures the expected time to mixing of the Markov chain corresponding to T. Given a strongly connected directed graph D, we consider the set ΣD of stochastic matrices whose directed graph is subordinate to D, and compute the minimum v ..."
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For an irreducible stochastic matrix T, the Kemeny constant K(T) measures the expected time to mixing of the Markov chain corresponding to T. Given a strongly connected directed graph D, we consider the set ΣD of stochastic matrices whose directed graph is subordinate to D, and compute the minimum value of K, taken over the set ΣD. The matrices attaining that minimum are also characterised, thus yielding a description of the transition matrices in ΣD that minimise the expected time to mixing. We prove that K(T) is bounded from above as T ranges over the irreducible members of ΣD if and only if D is an intercyclic directed graph, and in the case that D is intercyclic, we find the maximum value of K on the set ΣD. Throughout, our results are established using a mix of analytic and combinatorial techniques.