The pagerank vector: Properties, computation, approximation, and acceleration

SIAM Journal on Matrix Analysis and Applications

Volumn 28, Issue 2, 2006, Pages 551-575

  Brezinski, Claude ^a   Redivo Zaglia, Michela ^b  

^a UNIVERSITÉ LILLE 1   (France)

^b UNIVERSITY OF PADOVA   (Italy)

Author keywords

Convergence acceleration; Eigenvector computation; Pad approximation; PageRank; Power method; Web matrix

Indexed keywords

ALGORITHMS; COMPUTATIONAL METHODS; CONVERGENCE OF NUMERICAL METHODS; EIGENVALUES AND EIGENFUNCTIONS; MATRIX ALGEBRA; PROBLEM SOLVING;

CONVERGENCE ACCELERATION; EIGENVECTOR COMPUTATION; KRYLOV SUBSPACE METHOD; PAGERANK VECTOR; POWER METHOD; WEB MATRIX;

VECTORS;

EID: 34247383235     PISSN: 08954798     EISSN: 10957162     Source Type: Journal    
DOI: 10.1137/050626612     Document Type: Article

References (52)

1. A. ARASU, J. NOVAK, A. TOMKINS, AND J. TOMLIN, PageRank computation and the structure of the Web: Experiments and algorithms, in Proceedings of the Eleventh International World Wide Web Conference, ACM Press, New York, 2002; available online from http://www2002.org/CDROM/poster/173.pdf.
2. P. BERKHIN, A survey on PageRank computing. Internet Math., 2 (2005), pp. 73-120.
3. A. BERMAN AND R. J. PLEMMONS, Nonnegative Matrices in the Mathematical Sciences, SIAM, Philadelphia, 1994.
4. M. BIANCHINI, M. GORI, AND F. SCARSELLI, Inside Pagerank, ACM Trans. Internet Technol., 5 (2005), pp. 92-128.
5. P. BOLDI, M. SANTINI, AND S. VIGNA, PageRank as a function of the damping factor, in Proceedings of the Fourteenth International World Wide Web Conference, ACM Press, New York, 2005, pp. 557-566.
6. C. BREZINSKI, Padé-Type Approximation and General Orthogonal Polynomials, Internat. Ser. Numer. Math. 50, Birkhäuser Verlag, Basel, 1980.
7. C. BREZINSKI, A general extrapolation algorithm, Numer. Math., 35 (1980), pp. 175-187.
8. C. BREZINSKI, Projections Methods for Systems of Equations, North-Holland, Amsterdam, 1997.
9. C. BREZINSKI, Computational Aspects of Linear Control, Kluwer, Dordrecht, 2002.
10. C. BREZINSKI, Vector sequence transformations: Methodology and applications to linear systems, J. Comput. Appl. Math., 98 (1998), pp. 149-175.
11. C. BREZINSKI, Biorthogonal vector sequence transformations and Padé appropriation of vector series, Appl. Numer. Math., 41 (2002), pp. 437-442.
12. 2 d'Aitken, C. R. Acad. Sci. Paris Sér. A-B, 270 (1970), pp. 896-898.
13. C. BREZINSKI AND M. REDIVO-ZAGLIA, Extrapolation Methods. Theory and Practice, North-Holland, Amsterdam, 1991.
14. C. BREZINSKI AND M. REDIVO-ZAGLIA, Vector and matrix sequence transformations based on biorthogonality, Appl. Numer. Math., 21 (1996), pp. 353-373.
15. C. BREZINSKI AND M. REDIVO-ZAGLIA, New vector sequence transformations, Linear Algebra Appl., 389 (2004), pp. 189-213.
16. C. BREZINSKI AND M. REDIVO-ZAGLIA, Rational Extrapolation of the PageRank Vectors, in progress.
17. C. BREZINSKI, M. REDIVO-ZAGLIA, AND S. SERRA-CAPIZZANO, Extrapolation methods for PageRank computations, C. R. Math. Acad. Sci. Paris, 340 (2005), pp. 393-397.
18. S. BRIN AND L. PAGE, The anatomy of a large-scale hypertextual web search engine, Comput. Networks ISDN Syst., 30 (1998), pp. 107-117.
19. G. M. DEL CORSO, A. GULLI, AND F. ROMANI, Fast PageRank computation via a sparse linear system, in Algorithms and Models for the Web-Graph, S. Leonardi, ed., Lecture Notes in Comput. Sci. 3243, Springer-Verlag, New York, 2004, pp. 118-130.
20. L. ELDÉN, The Eigenvalues of the Google Matrix, Report LiTH-MAT-R-04-01, Department of Mathematics, Linköping University, Sweden, 2003.
21. F. R. GANTMACHER, The Theory of Matrices, Vol. 1, Chelsea, New York, 1959.
22. D. GLEICH, L. ZHUKOV, AND P. BERKHIN, Fast Parallel PageRank: A Linear System Approach, Technical report, Yahoo! Inc., 2004.
23. G. H. GOLUB AND C. GREIF, Arnoldi-Type Algorithms for Computing Stationary Distribution Vectors, with Application to PageRank, Technical report SCCM-04-15, Stanford University, Stanford, CA, 2004.
24. T. H. HAVELIWALA AND S. D. KAMVAR, The Second Eigenvalue of the Google Matrix, Technical report, Stanford University, Stanford, CA, 2003; available online from http://dbpubs.stanford.edu/pub/2003- 20.
25. I. C. F. IPSEN AND S. KIRKLAND, Convergence analysis of a PageRank updating algorithm by Langville and Meyer, SIAM J. Matrix Anal. Appl., 27 (2006), pp. 952-967.
26. I. C. F. IPSEN AND R. S. WILLS, Mathematical properties and analysis of Google's PageRank, Bol. Soc. Esp. Mat. Apl., 34 (2006), pp. 191-196.
27. K. JBILOU, A general projection algorithm for solving systems of linear equations, Numer. Algorithms, 4 (1993), pp. 361-377.
28. S. D. KAMVAR AND T. H. HAVELIWALA, The Condition Number of the PageRank Problem, Technical report, Stanford University, Stanford, CA, 2003; available online from http://dbpubs.stanford.edu/pub/2003- 36.
29. S. KAMVAR, T. HAVELIWALA, AND G. GOLUB, Adaptive methods for the computation of Page-Rank, Linear Algebra Appl., 386 (2004), pp. 51-65.
30. S. D. KAMVAR, T. H. HAVELIWALA, C. D. MANNING, AND G. H. GOLUB, Exploiting the Block Structure of the Web for Computing PageRank, Technical report, Stanford University, Stanford, CA, 2003; available online from http://dbpubs.stanford.edu/pub/2003-17.
31. S. D. KAMVAR, T. H. HAVELIWALA, C. D. MANNING, AND G. H. GOLUB, Extrapolations methods for accelerating PageRank computations, in Proceedings of the Twelfth International World Wide Web Conference, ACM Press, New York, 2003, pp. 261-270.
32. G. KOLLIAS, E. GALLOPOULOS, AND D. B. SZYLD, Asynchronous iterative computations with Web information retrieval structures: The PageRank case, in Proceedings of the Conference on Parallel Computing 2005, Malaga, Spain, 2005.
33. A. N. LANGVILLE AND C. D. MEYER, The use of linear algebra by web search engines, IMAGE Newsletter, 33 (2004), pp. 2-6.
34. A. N. LANGVILLE AND C. D. MEYER, Deeper inside PageRank, Internet Math., 1 (2005), pp. 335-400.
35. A. N. LANGVILLE AND C. D. MEYER, Updating PageRank with iterative aggregation, in Proceedings of the Thirteenth World Wide Web Conference, ACM Press, New York, 2004, pp. 392-393.
36. A. N. LANGVILLE AND C. D. MEYER, Google's PageRank and Beyond: The Science of Search Engine Rankings, Princeton University Press, Princeton, NJ, 2006.
37. C. P.-C. LEE. G. H. GOLUB, AND S. A. ZENIOS, A Fast Two-Stage Algorithm for Computing PageRank and Its Extensions, Technical report SCCM-2003-15, Scientific Computation and Computational Mathematics, Stanford University, Stanford, CA, 2003; available online from http://www-sccm.stanford.edu/nf-publications-tech.html.
38. A. C. MATOS, Acceleration results for the vector E-algorithm, Numer. Algorithms, 1 (1991), pp. 237-260.
39. A. C. MATOS, Convergence and acceleration properties for the vector ε-algorithm, Numer. Algorithms, 3 (1992), pp. 313-320.
40. C. D. MEYER, Matrix Analysis and Applied Linear Algebra, SIAM, Philadelphia, 2000.
41. L. PAGE, S. BRIN, R. MOTWANI, AND T. WINOGRAD, The PageRank Citation Ranking: Bringing Order to the Web, Technical report, Stanford University, Stanford, CA, 1999; available online from http://dbpubs.stanford.edu/pub/1999-66.
42. B. P. PUGACHEV, Acceleration of convergence of iterative processes and a method of solving systems of nonlinear equations, USSR Comput. Maths. Maths. Phys., 17 (1978), pp. 199-207.
43. M. REDIVO-ZAGLIA, Pseudo-Schur complements and their properties, Appl. Numer. Math., 50 (2004), pp. 511-519.
44. S. SERRA-CAPIZZANO, Jordan canonical form of the Google matrix: A potential contribution to the PageRank computation, SIAM J. Matrix Anal. Appl., 27 (2005), pp. 305-312.
45. A. SIDI, On a generalization of the Richardson extrapolation process, Numer. Math., 57 (1990), pp. 365-377.
46. A. SIDI, Approximation of Largest Eigenpairs of Matrices and Applications to PageRank Computation, Technical report CS-2004-16, Computer Science Department, Technion, Haifa, Israel, 2004.
47. W. J. STEWART, Introduction to the Numerical Solution of Markov Chains, Princeton University Press, Princeton, NJ, 1994.
48. J. VAN ISEGHEM, Vector Padé approximants, in Numerical Mathematics and Applications, R. Vichnevetsky and J. Vignes, eds., North-Holland, Amsterdam, 1986, pp. 73-77.
49. R. S. VARGA, Matrix Iterative Analysis, 2nd ed., Springer, New York, 2000.
50. YU. V. VOROBYEV, Method of Moments in Applied Mathematics, Gordon and Breach, New York, 1965.
51. P. WYNN, On the convergence and stability of the epsilon algorithm, SIAM J. Numer. Anal., 3 (1966), pp. 91-122.
52. F.-Z. ZHANG, ED., The Schur Complement and Its Applications, Springer, New York, 2005.