As a fundamental algorithm for collaborative processing over multi-agent systems, distributed consensus algorithm has been studied for optimizing its convergence rate. Due to the close analogy between the diffusion problem and the consensus algorithm, the previous trend in the literature is to transform the diffusion system from the spatially continuous domain into the spatially discrete one. In this transformation, the optimality is not necessarily preserved. In this paper, the reverse of this approach has been adopted, and it has been shown that the optimality can be preserved. This paper studies optimization of the Continuous-Time Consensus (CTC) problem on a weighted digraph with given average weight. Based on the detailed balance property, the CTC algorithm is converted into the weighted-average CTC algorithm. For the given distribution and average weight, a possible solution procedure has been provided. For finding the optimal weights corresponding to the weighted-average CTC algorithm with optimal convergence rate on a general graph. This solution procedure has been implemented based on the min-max theorem. For path topology, it is shown that the linearity of the drift term is the necessary and sufficient condition for the optimality of the consensus algorithm (and the corresponding diffusion system). Thus, the Pearson's class of discrete (continuous) distributions are optimal, where the closed-form formulas for the convergence rate, spectrum and other characteristics of the corresponding optimal consensus algorithm (diffusion system), i.e., the Hypergeometric types have been provided.

中文翻译：

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基于加权平均一致性的平流扩散系统优化


作为多智能体系统协同处理的基本算法，分布式一致性算法被研究以优化其收敛速度。由于扩散问题与共识算法的密切相似性，以往文献的趋势是将扩散系统从空间连续域转变为空间离散域。在此转换中，不一定保留最优性。在本文中，采用了与该方法相反的方法，并且表明可以保持最优性。本文研究了给定平均权重的加权有向图上的连续时间一致性（CTC）问题的优化。基于详细的余额特性，将CTC算法转换为加权平均CTC算法。对于给定的分布和平均重量，提供了可能的求解过程。用于在一般图上找到具有最佳收敛速度的加权平均CTC算法对应的最佳权重。该求解过程是基于最小-最大定理实现的。对于路径拓扑，表明漂移项的线性是共识算法（以及相应的扩散系统）最优的充要条件。因此，皮尔逊类离散（连续）分布是最优的，其中已经提供了相应最优一致性算法（扩散系统）的收敛速度、谱和其他特性的封闭式公式，即超几何类型。