  This paper proposes a new distributed algorithm for solving linear systems
associated with a sparse graph under a generalised diagonal dominance
assumption. The algorithm runs iteratively on each node of the graph, with low
complexities on local information exchange between neighbouring nodes, local
computation and local storage. For an acyclic graph under the condition of
diagonal dominance, the algorithm is shown to converge to the correct solution
in a finite number of iterations, equalling the diameter of the graph. For a
loopy graph, the algorithm is shown to converge to the correct solution
asymptotically. Simulations verify that the proposed algorithm significantly
outperforms the classical Jacobi method and a recent distributed linear system
solver based on average consensus and orthogonal projection.
