Title: A Distributed Algorithm for Solving Linear Algebraic Equations Over Random Networks,
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.