  We study the approach to obtaining least squares solutions to systems of
linear algebraic equations over networks by using distributed algorithms. Each
node has access to one of the linear equations and holds a dynamic state. The
aim for the node states is to reach a consensus as a least squares solution of
the linear equations by exchanging their states with neighbors over an
underlying interaction graph. A continuous-time distributed least squares
solver over networks is developed in the form of the famous Arrow-Hurwicz-Uzawa
flow. A necessary and sufficient condition is established on the graph
Laplacian for the continuous-time distributed algorithm to give the least
squares solution in the limit, with an exponentially fast convergence rate. The
feasibility of different fundamental graphs is discussed including path graph,
star graph, etc. Moreover, a discrete-time distributed algorithm is developed
by Euler's method, converging exponentially to the least squares solution at
the node states with suitable step size and graph conditions. The exponential
convergence rate for both the continuous-time and discrete-time algorithms
under the established conditions is confirmed by numerical examples. Finally,
we investigate the performance of the proposed flow under switching networks,
and surprisingly, switching networks at high switching frequencies can lead to
approximate least square solvers even if all graphs in the switching signal
fail to do so in the absence of structure switching.
