Recurrent Averaging Inequalities in Multi-Agent Control and Social Dynamics Modeling

TL;DR

This paper introduces a unified analytical framework using recurrent averaging inequalities to study the convergence properties of various multi-agent control algorithms based on iterative averaging.

Contribution

It develops a novel theory of recurrent averaging inequalities and applies it to analyze the convergence of multiple multi-agent algorithms in a unified manner.

Findings

Unified analysis of multi-agent algorithms using RAIs

Convergence conditions derived for various algorithms

Simplified understanding of averaging-based control methods

Abstract

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.