Convergence Speed of Binary Interval Consensus

Abstract

We consider the convergence time for solving the binary consensus problem using the interval consensus algorithm proposed by Bénézit, Thiran, and Vetterli [Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP), IEEE Press, Piscataway, NJ, 2009, pp. 3661–3664]. In the binary consensus problem, each node initially holds one of two states and the goal for each node is to correctly decide which one of these two states was initially held by a majority of nodes. We derive an upper bound on the expected convergence time that holds for arbitrary connected graphs, which is based on the location of eigenvalues of some contact rate matrices. We instantiate our bound for particular networks of interest, including complete graphs, paths, cycles, star-shaped networks, and Erdös–Rényi random graphs; for these graphs, we compare our bound with alternative computations. We find that for all these examples our bound is tight, yielding the exact order with respect to the number of nodes. We pinpoint the fact that the expected convergence time critically depends on the voting margin defined as the difference between the fraction of nodes that initially held the majority and the minority states, respectively. The characterization of the expected convergence time yields an exact relation between the expected convergence time and the voting margin for some of these graphs, which reveals how the expected convergence time goes to infinity as the voting margin approaches zero. Our results provide insight into how the expected convergence time depends on the network topology, which can be used for performance evaluation and network design. The results are of interest in the context of networked systems, in particular, peer-to-peer networks, sensor networks, and distributed databases.