Convergence Speed of Binary Interval Consensus

We consider the convergence time for solving the binary interval consensus problem using a distributed algorithm proposed by Benezit at al (2009) for computing the quantized average value. 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 the 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, star-shaped networks, and Erdos-Renyi random graphs, and in the former two cases compare with alternative computations. We find that for all these examples our bound is of 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 the nodes that initially held the majority and the minority states, respectively. We derive an exact relation between the expected convergence time and the voting margin, for some of these graphs, which reveals how the expected convergence time tends to infinity as the voting margin approaches zero. Our results provide insights on 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 peer-to-peer systems; in particular, for sensor networks and distributed databases.