A Distributed Algorithm for Solving Linear Algebraic Equations Over Random Networks

In this paper, the problem of solving linear algebraic equations of the form $Ax=b$ among multi agents is considered. It is assumed that the interconnection graphs over which the agents communicate are random. It is assumed that each agent only knows a subset of rows of the partitioned matrix $[A,\ b]$. The problem is formulated such that this formulation does not require distribution dependency of random communication graphs. The random Krasnoselskii-Mann iterative algorithm is applied for almost sure convergence to a solution of the problem for any matrices $A$ and $b$ and any initial conditions of agents' states. The algorithm converges almost surely independently from the distribution and, therefore, is amenable to completely asynchronous operations withot B-connectivity assumption. Based on initial conditions of agents' states, we show that the limit point of the sequence generated by the algorithm is determined by the unique solution of a convex optimization problem independent from the distribution of random communication graphs.