Many multi-agent control algorithms and dynamic agent-based models arising in natural and social sciences are based on the principle of iterative averaging. Each agent is associated to a value of interest, which may represent, for instance, the opinion of an individual in a social group, the velocity vector of a mobile robot in a flock, or the measurement of a sensor within a sensor network. This value is updated, at each iteration, to a weighted average of itself and of the values of the adjacent agents. It is well known that, under natural assumptions on the network’s graph connectivity, this local averaging procedure eventually leads to global consensus, or synchronization of the values at all nodes. Applications of iterative averaging include, but are not limited to, algorithms for distributed optimization, for solution of linear and nonlinear equations, for multi-robot coordination and for opinion formation in social groups. Although these algorithms have similar structures, the mathematical techniques used for their analysis are diverse, and conditions for their convergence and differ from case to case. In this paper, we review many of these algorithms and we show that their properties can be analyzed in a unified way by using a novel tool based on recurrent averaging inequalities (RAIs). We develop a theory of RAIs and apply it to the analysis of several important multi-agent algorithms recently proposed in the literature.

自然科学和社会科学中出现的许多多智能体控制算法和基于动态智能体的模型都是基于迭代平均的原理。每个代理都与一个感兴趣的值相关联，该感兴趣的值可以表示例如社交团体中某人的意见，羊群中移动机器人的速度矢量或传感器网络中传感器的测量值。在每次迭代中，此值都会更新为其自身和相邻代理的值的加权平均值。众所周知，在对网络图连接性的自然假设下，这种局部平均过程最终会导致全局共识或所有节点值的同步。迭代平均的应用包括但不限于用于分布式优化，线性和非线性方程式求解，多机器人协调以及社会群体意见形成的算法。尽管这些算法具有相似的结构，用于它们的分析的数学技术是多种多样的，并且它们收敛的条件也因情况而异。在本文中，我们回顾了许多这些算法，并表明可以使用基于递归平均不等式（RAI）的新颖工具以统一的方式分析其属性。我们发展了RAI的理论，并将其应用于最近在文献中提出的几种重要的多主体算法的分析。