Finite-time and Asymptotic Convergence of Distributed Averaging and Maximizing Algorithms

In this paper, we investigate a generalized consensus algorithm which unifies several distributed averaging and maximizing algorithms considered in the literature. Each node iteratively updates its state as a weighted average of the minimum and maximum values of its neighbors. Necessary and sufficient conditions are presented on the communication graph to ensure global consensus. For time-dependent graphs, we show that quasi-strong connectivity is critical for averaging, as is strong connectivity for maximizing. For state-dependent graphs, we consider a $\mu$-nearest-neighbor rule, in which each node interacts with its $\mu$ nearest smaller neighbors ($\mu$ nodes with smaller state values), and the $\mu$ nearest larger neighbors. Under such state-dependent updates, we show that the averaging algorithm leads to asymptotic consensus, while the maximizing algorithm leads to finite-time consensus. Moreover, for the averaging case, it is proven that $2\mu+1$ is a critical number of nodes for finite-time convergence, as finite-time convergence is almost never achieved if the graph has more than $2\mu+1$ nodes. The results characterize some fundamental similarity and difference between distributed averaging and maximizing algorithms.