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In this paper, we formulate and investigate a generalized consensus algorithmwhich makes an attempt to unify distributed averaging and maximizing algorithmsconsidered in the literature. Each node iteratively updates its state as atime-varying weighted average of its own state, the minimal state, and themaximal state of its neighbors. We prove that finite-time consensus is almostimpossible for averaging under this uniform model. Both time-dependent andstate-dependent graphs are considered, and various necessary and/or sufficientconditions are presented on the consensus convergence. For time-dependentgraphs, we show that quasi-strong connectivity is critical for averaging, as isstrong connectivity for maximizing. For state-dependent graphs defined by a$\mu$-nearest-neighbor rule, where each node interacts with its $\mu$ nearestsmaller neighbors and the $\mu$ nearest larger neighbors, we show that $\mu+1$is a critical threshold on the total number of nodes for the transit fromfinite-time to asymptotic convergence for averaging, in the absence of nodeself-confidence. The threshold is $2\mu$ if each node chooses to connect onlyto neighbors with unique values. Numerical examples illustrate the tightness ofthe conditions. The results characterize some fundamental similarities anddifferences between distributed averaging and maximizing algorithms.

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