A Distributed Algorithm for Solving Linear Algebraic Equations Over Random Networks

TL;DR

This paper introduces a distributed algorithm for solving linear equations over random networks, capable of handling asynchronous updates and unreliable communications, with proven convergence to the solution.

Contribution

It presents a novel distributed algorithm that converges almost surely for solving linear equations over random, unreliable network topologies without requiring graph distribution knowledge.

Findings

Algorithm converges almost surely and in mean square.

Convergence point is the unique solution of a convex optimization problem.

Rate of convergence cannot be guaranteed.

Abstract

In this paper, we consider the problem of solving linear algebraic equations of the form $Ax=b$ among multi agents which seek a solution by using local information in presence of random communication topologies. The equation is solved by $m$ agents where each agent only knows a subset of rows of the partitioned matrix $[A,b]$. We formulate the problem such that this formulation does not need the distribution of random interconnection graphs. Therefore, this framework includes asynchronous updates or unreliable communication protocols without B-connectivity assumption. We apply the random Krasnoselskii-Mann iterative algorithm which converges almost surely and in mean square to a solution of the problem for any matrices $A$ and $b$ and any initial conditions of agents' states. We demonestrate that the limit point to which the agents' states converge is determined by the unique solution of a convex optimization problem regardless of the distribution of random communication graphs. Eventually, we show by two numerical examples that the rate of convergence of the algorithm cannot be guaranteed.