An Upper Bound on the Convergence Time for Quantized Consensus of Arbitrary Static Graphs

Abstract

We analyze a class of distributed quantized consensus algorithms for arbitrary static networks. In the initial setting, each node in the network has an integer value. Nodes exchange their current estimate of the mean value in the network, and then update their estimation by communicating with their neighbors in a limited capacity channel in an asynchronous clock setting. Eventually, all nodes reach consensus with quantized precision. We analyze the expected convergence time for the general quantized consensus algorithm proposed by Kashyap et al. (“Quantized consensus,” Automatica, 2007). We use the theory of electric networks, random walks, and couplings of Markov chains to derive an O(N <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> log N) upper bound for the expected convergence time on an arbitrary graph of size N, improving on the state of art bound of O(N <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">5</sup> ) for quantized consensus algorithms. Our result is not dependent on graph topology. Example of complete graphs is given to show how to extend the analysis to graphs of given topology. This is consistent with the analysis in “Convergence speed of binary interval consensus,” (M. Draief and M. Vojnovic, SIAM J. Control and Optimization, 2012.