A Note on the Convergence of SOR for the PageRank Problem

A curious phenomenon when it comes to solving the linear system formulation of the PageRank problem is that while the convergence rate of Gauss-Seidel shows an improvement over Jacobi by a factor of approximately two, successive overrelaxation (SOR) does not seem to offer a meaningful improvement over Gauss-Seidel. This has been observed experimentally and noted in the literature, but to the best of our knowledge there has been no analytical explanation for this thus far. This convergence behavior is surprising because there are classes of matrices for which Gauss-Seidel is faster than Jacobi by a similar factor of two, and SOR accelerates convergence by an order of magnitude compared to Gauss-Seidel. In this short paper we prove analytically that the PageRank model has the unique feature that there exist PageRank linear systems for which SOR does not converge outside a very narrow interval depending on the damping factor, and that in such situations Gauss-Seidel may be the best choice among the relaxation parameters. Conversely, we show that within that narrow interval, there exists no PageRank problem for which SOR does not converge. Our result may give an analytical justification for the popularity of Gauss-Seidel as a solver for the linear system formulation of PageRank.