The consensus problem in distributed computing involves a network of agents aiming to compute the average of their initial vectors through local communication, represented by an undirected graph. This paper focuses on the studying of this problem using an average-case analysis approach, particularly over regular graphs. Traditional algorithms for solving the consensus problem often rely on worst-case performance evaluation scenarios, which may not reflect typical performance in real-world applications. Instead, we apply average-case analysis, focusing on the expected spectral distribution of eigenvalues to obtain a more realistic view of performance. Key contributions include deriving the optimal method for consensus on regular graphs, showing its relation to the Heavy Ball method, analyzing its asymptotic convergence rate, and comparing it to various first-order methods through numerical experiments.

翻译：分布式计算中的共识问题涉及一个由智能体组成的网络，其目标是通过局部通信计算各自初始向量的平均值，该网络由无向图表示。本文重点采用平均情况分析方法研究该问题，特别是在正则图上的应用。解决共识问题的传统算法通常依赖于最坏情况性能评估场景，这可能无法反映实际应用中的典型性能。相反，我们采用平均情况分析，重点关注特征值的期望谱分布，以获得更符合实际的性能视角。主要贡献包括推导出正则图上共识问题的最优方法，展示其与重球法的关联，分析其渐近收敛速率，并通过数值实验将其与多种一阶方法进行比较。