Convergence analysis of a PageRank updating algorithm by Langville and Meyer

SIAM Journal on Matrix Analysis and Applications

Volumn 27, Issue 4, 2006, Pages 952-967

  Ipsen, Ilse C F ^a   Kirkland, Steve ^b  

^a NORTH CAROLINA STATE UNIVERSITY   (United States)

^b UNIVERSITY OF REGINA   (Canada)

Author keywords

Aggregation disaggregation; Google; Markov chain; PageRank; Power method; Stochastic complement

Indexed keywords

AGGREGATION/DISAGGREGATION; GOOGLE; PAGERANK; POWER METHOD; STOCHASTIC COMPLEMENT;

ALGORITHMS; ASYMPTOTIC STABILITY; COMPUTATION THEORY; DIGITAL ARITHMETIC; ITERATIVE METHODS; MARKOV PROCESSES; SEARCH ENGINES;

CONVERGENCE OF NUMERICAL METHODS;

EID: 33646455491     PISSN: 08954798     EISSN: 10957162     Source Type: Journal    
DOI: 10.1137/S0895479804439808     Document Type: Conference Paper

References (30)

1. A. BERMAN AND R. J. PLEMMONS, Nonnegative Matrices in the Mathematical Sciences, Classics Appl. Math., SIAM, Philadelphia, 1994.
2. C. BREZINSKI AND M. REDIVO-ZAGLIA, The PageRank vector: Properties, computation, approximation and acceleration, SIMAX, submitted.
3. C. BREZINSKI, M. REDIVO-ZAGLIA, AND S. SERRA-CAPIZZANO, Extrapolation Methods for PageRank Computations, Comptes Rendus Acad. Sci., Sér. I, 340 (2005), pp. 393-397.
4. S. BRIN AND L. PAGE, The anatomy of a large-scale hypertextual web search engine, Comput. Networks and ISDN Systems, 30 (1998), pp. 107-117.
5. R. W. COTTLE, Manifestations of the Schur complement, Linear Algebra Appl., 8 (1974), pp. 189-211.
6. T. H. HAVELIWALA AND S. D. KAMVAR, The Second Eigenvalue of the Google Matrix, Technical report 2003-20, Stanford University, Stanford, CA, 2003, http://dbpubs.stanford.edu/pub/2003-20.
7. T. H. HAVELIWALA, S. D. KAMVAR, D. KLEIN, C. D. MANNING, AND G. H. GOLUB, Computing PageRank Using Power Extrapolation, Technical report 2003-45, Stanford University, Stanford, CA, 2003, http://dbpubs.stanford.edu/pub/2003-45.
8. C. R. JOHNSON AND C. XENOPHONTOS, Irreducibility and primitivity of Perron complements: Application of the compressed directed graph, in Graph Theory and Sparse Matrix Computation, IMA Vol. Math. Appl. 56, A. George, J. R. Gilbert, and J. W. H. Liu, eds., Springer, New York, 1993, pp. 101-106.
9. S. D. KAMVAR, T. H. HAVELIWALA, AND G. H. GOLUB, Adaptive methods for the computation of PageRank, Linear Algebra Appl., 386 (2004), pp. 51-65.
10. 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 SCCM03-02, Stanford University, Stanford, CA, 2003, http://www-sccm. stanford.edu/nf-publications-tech.html.
11. S. D. KAMVAR, T. H. HAVELIWALA, C. D. MANNING, AND G. H. GOLUB, Extrapolation methods for accelerating PageRank computations, in Proceedings of the Twelfth International World Wide Web Conference (WWW2003), Toronto, 2003, ACM Press, New York, pp. 261-270.
12. A. N. LANGVILLE AND C. D. MEYER, Updating PageRank Using the Group Inverse and Stochastic Complementation, Technical report CRSC-TR02-32, North Carolina State University, Raleigh, NC, 2002, http://www.ncsu.edu/crsc/reports/ reports02.html.
13. A. N. LANGVILLE AND C. D. MEYER, Updating the Stationary Vector of an Irreducible Markov Chain, Technical report CRSC-TR02-33, North Carolina State University, Raleigh, NC, 2002, http://www.ncsu.edu/crsc/reports/reports02.html.
14. A. N. LANGVILLE AND C. D. MEYER, Fiddling with PageRank, Technical report CRSC-TR03-34, North Carolina State University, Raleigh, NC, 2003, http://www.ncsu.edu/crsc/reports/reports03.html.
15. A. N. LANGVILLE AND C. D. MEYER, Updating the Stationary Vector of an Irreducible Markov Chain with an Eye on Google's PageRank, Technical report, North Carolina State University, Raleigh, NC, 2004.
16. C. P.-C. LEE, G. H. GOLUB, AND S. A. ZENIOS, A Fast Two-Stage Algorithm for Computing PageRank, Technical report SCCM03-15, Stanford University, Stanford, CA, 2003, http://www-sccm.stanford.edu/nf-publications-tech.html.
17. I. MAREK AND P, MAYER, Convergence theory of some classes of iterative aggregation/disaggregation methods for computing stationary probability vectors of stochastic matrices, Linear Algebra Appl., 363 (2003), pp. 177-200.
18. I. MAREK AND D. B. SZYLD, Local convergence of the (exact and inexact) iterative aggregation method for linear systems and Markov operators, Numer. Math., 69 (1994), pp. 61-82.
19. C. D. MEYER, Stochastic complementation, uncoupling Markov chains, and the theory of nearly reducible systems, SIAM Rev., 31 (1989), pp. 240-272.
20. C. D. MEYER, Uncoupling the Perron eigenvector problem, Linear Algebra Appl., 114/115 (1989), pp. 69-94.
21. C. B. MOLER, The world's largest matrix computation, MATLAB News and Notes, October, 2002.
22. C. B. MOLER, Numerical Computing with MATLAB, SIAM, Philadelphia, 2004.
23. M. NEUMANN AND J. XU, On the stability of the computation of the stationary probabilities of Markov chains using Perron complements, Numer. Linear Algebra Appl., 10 (2003), pp. 603-618.
24. A. Y. NG, A. X. ZHENG, AND M. I. JORDAN, Link analysis, eigenvectors and stability, in Proceedings of the 17th International Joint Conference on Artificial Intelligence (IJCAI), B. Nabel, ed., Morgan Kaufmann/Elsevier, San Francisco, 2001, pp. 903-910.
25. L. PAGE, S. BRIN, R. MOTWANI, AND T. WINOGRAD, The PageRank Citation Ranking: Bringing Order to the Web, 1999, http://dbpubs.stanford.edu/pub/1999- 66.
26. E. SENETA, Non-negative Matrices and Markov Chains, Springer-Verlag, New York, 1981.
27. E. SENETA, Explicit forms for ergodicity coefficients and spectrum localization, Linear Algebra Appl., 60 (1984), pp. 187-197.
28. 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.
29. W. J. STEWART, Introduction to the Numerical Solution of Markov Chains, Princeton University Press, Princeton, NJ, 1994.
30. R. S. VARGA, Matrix Iterative Analysis, Prentice Hall, Englewood Cliffs, NJ, 1962.