  In this paper, we propose distributed solvers for systems of linear equations
given by symmetric diagonally dominant M-matrices based on the parallel solver
of Spielman and Peng. We propose two versions of the solvers, where in the
first, full communication in the network is required, while in the second
communication is restricted to the R-Hop neighborhood between nodes for some $R
\geq 1$. We rigorously analyze the convergence and convergence rates of our
solvers, showing that our methods are capable of outperforming state-of-the-art
techniques.
  Having developed such solvers, we then contribute by proposing an accurate
distributed Newton method for network flow optimization. Exploiting the
sparsity pattern of the dual Hessian, we propose a Newton method for network
flow optimization that is both faster and more accurate than state-of-the-art
techniques. Our method utilizes the distributed SDDM solvers for determining
the Newton direction up to any arbitrary precision $\epsilon >0$. We analyze
the properties of our algorithm and show superlinear convergence within a
neighborhood of the optimal. Finally, in a set of experiments conducted on
randomly generated and barbell networks, we demonstrate that our approach is
capable of significantly outperforming state-of-the-art techniques.
