  We study distributed network flows as solvers in continuous time for the
linear algebraic equation $\mathbf{z}=\mathbf{H}\mathbf{y}$. Each node $i$ has
access to a row $\mathbf{h}_i^{\rm T}$ of the matrix $\mathbf{H}$ and the
corresponding entry $z_i$ in the vector $\mathbf{z}$. The first "consensus +
projection" flow under investigation consists of two terms, one from standard
consensus dynamics and the other contributing to projection onto each affine
subspace specified by the $\mathbf{h}_i$ and $z_i$. The second "projection
consensus" flow on the other hand simply replaces the relative state feedback
in consensus dynamics with projected relative state feedback. Without
dwell-time assumption on switching graphs as well as without positively lower
bounded assumption on arc weights, we prove that all node states converge to a
common solution of the linear algebraic equation, if there is any. The
convergence is global for the "consensus + projection" flow while local for the
"projection consensus" flow in the sense that the initial values must lie on
the affine subspaces. If the linear equation has no exact solutions, we show
that the node states can converge to a ball around the least squares solution
whose radius can be made arbitrarily small through selecting a sufficiently
large gain for the "consensus + projection" flow under fixed bidirectional
graphs. Semi-global convergence to approximate least squares solutions is
demonstrated for general switching directed graphs under suitable conditions.
It is also shown that the "projection consensus" flow drives the average of the
node states to the least squares solution with complete graph. Numerical
examples are provided as illustrations of the established results.
