An upper bound on the convergence time for quantized consensus

Abstract

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(N 3 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 4 log N ) for binary consensus and O(N 5 ) 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.

Fast convergence of quantized consensus using Metropolis chains

“…Using Theorem 2 in the last relation of (4), we obtain that the expected time until all walkers collapse to one big clump is of the order of at most O(n 2 log n). Finally, using the same argument as in Corollary 4 in [16] we can see that O(n 2 log n) is an upper bound for the expected convergence time of the quantized Metropolis dynamics.…”

“…Moreover, an upper bound of O(n 2 m max D max log 2 n) on the expected convergence time of unbiased quantized consensus in the presence of connected time-varying networks has been given in [15], where m max and D max denote, respectively, the maximum number of edges and the maximum diameter in the sequences of time-varying networks. The biased quantized consensus problem was addressed in [16], where it was shown that the expected convergence time in general networks is at most O(n 3 log n). Moreover, a deterministic version of the quantized-consensus algorithm was introduced in [17], where the authors showed a fast convergence time of O(n 2 ) for such dynamics.…”

Convergence time for unbiased quantized consensus

“…As in [9] and [6], we define the potential function Φ(·, ·) : V × V → R to be Φ(x, y) = H Z (x, y) + H Z (y, w) − H Z (w, y). Now consider the original joint process for two walkers on a finite graph.…”

“…Since by definitionT (G) is independent of the initial states of walkers, max x,y M (x, y) ≤ T (G). For the upper bound, we use the same argument used in [6]. Indeed half of the time when two walkers meet in the virtual process, they do not meet in the original process and they stay in the same position.…”

“…However, depending on the initial profile and the distribution of choosing the edges at each time instant, the convergence time in expectation may vary. Studies on the behavior of the quantized consensus algorithm based on natural random walks and biased random walks can be found in [5] and [6]. In this paper, we consider the quantized consensus algorithm when each edge has equal probability of being chosen at each time step (unbiased).…”