Scalable, Distributed Algorithms for Solving Linear Equations via Double-Layered Networks

Abstract

This paper proposes scalable, distributed algorithms for solving linear equations by integrating two mechanisms, termed consensus and conservation, in double-layered multiagent networks. The multiagent network considered in this paper is composed of clusters and each cluster consists of an aggregator and a subnetwork of agents. By achieving consensus and conservation through agent–agent communications in the same cluster and aggregator–aggregator communications among different clusters, respectively, distributed algorithms are devised for agents to cooperatively achieve a solution to the overall linear equation. These algorithms outperform existing algorithms, including but not limited to the following aspects—first, each agent does not have to know as much as a complete row or column of the overall equation; second, each agent only needs to control as few as two scalar states when the number of clusters and the number of agents are sufficiently large; third, the dimensions of agents’ states in the proposed algorithms do not have to be the same (while in contrast, algorithms based on the idea of standard consensus inherently require all agents’ states to be of the same dimension). Both analytical proof and simulation results are provided to validate exponential convergence of the proposed distributed algorithms in solving linear equations.