excellent question! Basically you could use the SDD solver http://math.mit.edu/~kelner/Publications/Docs/1301.6628v1.pd... in any context where you'd use an SVD (pseudo inverse / least squares) solver, and your matrix is symmetric and diagonally dominant.

The key bit however, is that the SDD solver as above, has really nice asymptotics for sparse matrices.

theres a few fancy algorithms that need to be engineered along the way, and theres also the question about how the constant factors work out in practice!