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We analyze a class of distributed quantized consensus algorithms for arbitrary 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 start the analysis with a special case of a distributed binary voting algorithm, then proceed to the expected convergence time for the general quantized consensus algorithm proposed by Kashyap et al. We use the theory of electric networks, random walks, and couplings of Markov chains to derive an O(N3 logN) upper bound for the expected convergence time on an arbitrary graph of size N, improving on the state of art bound of O(N4 logN) for binary consensus and O(N5) for quantized consensus algorithms. Our result is not dependent on graph topology. Simulations on special graphs such as star networks, line graphs, lollipop graphs, and Erdös-Rényi random graphs are performed to validate the analysis. This work has applications to load balancing, coordination of autonomous agents, estimation and detection, decision-making networks, peer-to-peer systems, etc