In this talk, motivated by the application in computing distance metric for protein interaction networks, we will discuss the algorithmic development of fast solvers for graph Laplacian systems.  Two different solvers will be introduced.  One solver is based on the algebraic multigrid method and the other one is based on a special subspace correction method.  We will discuss their theoretical convergence and computational complexity.  Finally, applications in protein interaction networks will be presented and possible generalization will be discussed.