A Preconditioned and Shifted GMRES Algorithm for the PageRank Problem with Multiple Damping Factors

Google has become one of the most popular and successful search engines in recent years. Google's success can be attributed to its simple and elegant algorithm: PageRank. In practice, one often needs to solve the PageRank problem with multiple damping factors or with multiple damping factors and multiple personalization vectors. The conventional PageRank algorithm has to solve these problems one by one. The shifted GMRES$(m)$ algorithm can be used to solve them in the same search subspace. However, there are two disadvantages to this algorithm. The first is “near singularity,” and the second is “stagnation.” In this paper, we first present a modified and shifted GMRES$(m)$ algorithm to deal with the problem of near singularity. In order to overcome the drawback of stagnation and to improve convergence, we propose a polynomial preconditioner for the modified algorithm. We show that the resulting algorithm can circumvent the drawbacks of near singularity and stagnation that occur in its original counterpart...

[1]  Ramesh Govindan,et al.  Making Eigenvector-Based Reputation Systems Robust to Collusion , 2004, WAW.

[2]  Gene H. Golub,et al.  Matrix computations (3rd ed.) , 1996 .

[3]  Denis Vanderstraeten,et al.  A Stable and Efficient Parallel Block Gram-Schmidt Algorithm , 1999, Euro-Par.

[4]  R. Freund.  Solution of shifted linear systems by quasi-minimal residual iterations , 1993 .

[5]  R. Morgan,et al.  Deflated GMRES for systems with multiple shifts and multiple right-hand sides☆ , 2007, 0707.0502.

[6]  Gene H. Golub,et al.  Extrapolation methods for accelerating PageRank computations , 2003, WWW '03.

[7]  G. Golub,et al.  An Arnoldi-type algorithm for computing page rank , 2006 .

[8]  Joseph F. Grcar,et al.  Operator Coefficient Methods for Linear Equations , 2012, 1203.2390.

[9]  David F. Gleich,et al.  Random Alpha PageRank , 2009, Internet Math..

[10]  Rajeev Motwani,et al.  The PageRank Citation Ranking : Bringing Order to the Web , 1999, WWW 1999.

[11]  Gang Wu,et al.  Arnoldi versus GMRES for computing pageRank: A theoretical contribution to google's pageRank problem , 2010, TOIS.

[12]  Valeria Simoncini,et al.  New conditions for non-stagnation of minimal residual methods , 2008, Numerische Mathematik.

[13]  Dianne P. O'Leary,et al.  Complete stagnation of gmres , 2003 .

[14]  Yousef Saad,et al.  Iterative methods for sparse linear systems , 2003 .

[15]  Amy Nicole Langville,et al.  Google's PageRank and beyond - the science of search engine rankings , 2006 .

[16]  Andreas Frommer,et al.  Restarted GMRES for Shifted Linear Systems , 1998, SIAM J. Sci. Comput..

[17]  Valeria Simoncini,et al.  Recent computational developments in Krylov subspace methods for linear systems , 2007, Numer. Linear Algebra Appl..

[18]  Gene H. Golub,et al.  Matrix computations , 1983 .

[19]  David F. Gleich,et al.  An Inner-Outer Iteration for Computing PageRank , 2010, SIAM J. Sci. Comput..

[20]  H. Elman.  Iterative methods for large, sparse, nonsymmetric systems of linear equations , 1982 .

[21]  Taher H. Haveliwala.  Topic-Sensitive PageRank: A Context-Sensitive Ranking Algorithm for Web Search , 2003, IEEE Trans. Knowl. Data Eng..