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About the Power Law of the PageRank Vector Component Distribution. Part 1. Numerical Methods for Finding the PageRank Vector

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Abstract

In Part 1 of this paper we consider the web-page ranking problem, also known as the problem of finding the PageRank vector, or the Google problem. We discuss the link between this problem and the ergodic theorem and describe different numerical methods to solve this problem together with their theoretical background, such asMarkov chain Monte Carlo and equilibrium in a macrosystem.

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References

  1. Agaev, R.P. and Chebotarev, P.Yu., Convergence and Stability in Problems of Agreement of Characteristics (Overview of the Baseline Results), Upravl. Bol’sh. Sist., 2010, vol. 30, no. 1, pp. 470–505.

    Google Scholar 

  2. Anikin, A.S., Gasnikov, A.V., Gornov, A.Yu., Kamzolov, D.I., Maksimov, Yu.V., and Nesterov, Yu.E., Effective Numerical Methods for Solving the Page Rank Problem for Doubly Sparse Matrices, Trudy MFTI, 2015, vol. 7, no. 4, pp. 74–94.

    Google Scholar 

  3. Arnol’d, V.I., Tsepnye drobi (Continued Fractions), Moscow: MTSNMO, 2001.

    Google Scholar 

  4. Baymurzina, D.R., Gasnikov, A.V., and Gasnikova, E.V., The Theory of Macrosystems in Terms of Stochastic Chemical Kinetics, Trudy MFTI, 2015, vol. 7, no. 4, pp. 95–103.

    Google Scholar 

  5. Batishcheva, Ya.G. and Vedenyapin, V.V., The Second Law of Thermodynamics for Chemical Kinetics, Mat. Mod., 2005, vol. 17, no. 8, pp. 106–110.

    MathSciNet  MATH  Google Scholar 

  6. Blank, M.L., Ustoichivost’ i lokalizatsiya v haoticheskoi dinamike (Stability and Localization in Chaotic Dynamics), Moscow: MTSNMO, 2001.

    Google Scholar 

  7. Bogdanov, K.Yu., Dynamics of Social Inequality, Kvant, 2004, no. 5, pp. 7–12.

    Google Scholar 

  8. Bogdanov, K.Yu., Predator and Prey: The Equation of Coexistence, Kvant, 2014, nos. 4/5, pp. 13–17.

    Google Scholar 

  9. Buzun, N.O., Gasnikov, A.V., Goncharov, F.O., Gorbachev, O.G., Guz, S.A., Krymova, E.A., Natan, A.A., and Chernousova, E.O., Stokhasticheskii analiz v zadachakh, chast’ 1 (Stochastic Analysis in Problems. Part 1), Moscow: Izd. MIPT, 2016.

    Google Scholar 

  10. Vaydlih, V., Sotsiodinamika: sistemnyi podkhod k matematicheskomu modelirovaniyu v sotsial’nykh naukakh (Socio-Dynamics: A Systematic Approach to Mathematical Modeling in the Social Sciences), Moscow: URSS, 2010.

    Google Scholar 

  11. Gal’perin, G.A. and Zemlyakov, A.N., Matematicheskie bil’yardy (Mathematical Billiards), Moscow: Nauka, 1990.

    MATH  Google Scholar 

  12. Gardiner, K.V., Stokhasticheskie metody v estestvennykh naukakh (Stochastic Methods in the Natural Sciences), Moscow: Mir, 1986.

    MATH  Google Scholar 

  13. Gasnikov, A.V., Vvedenie v matematicheskoe modelirovanie transportnykh potokov (Introduction to Mathematical Modeling of Transport Flows), 2nd ed., Moscow: MTSNMO, 2013.

    Google Scholar 

  14. Gasnikov, A.V., Effektivnye chislennye metody poiska ravnovesii v bol’shikh transportnykh setyakh (Effective Numerical Methods of Searching for Equilibria in Large Transport Systems), Doctoral (Phys.- Math) Dissertation, Moscow: MTSNMO, 2016.

    Google Scholar 

  15. Gasnikov, A.V. and Gasnikova, E.V., On Entropy-Like Functionals Arising in Stochastic Chemical Dynamics in Concentration of Invariant Measure and as the Lyapunov Functions of the Dynamics of Quasi-Averages, Mat. Zametki, 2013, vol. 94, no. 6, pp. 816–824.

    Article  MATH  Google Scholar 

  16. Gasnikov, A.V., Gasnikova, E.V., Mendel’, M.A., and Chepurchenko, K.V., Evolutionary Conclusions of Entropy Model for Calculating O-D Matrix, Mat. Model., 2016, vol. 28, no. 4, pp. 111–124.

    MathSciNet  MATH  Google Scholar 

  17. Gasnikov, A.V. and Dmitriev, D.Yu., About Effective Randomized Algorithms for Finding the Page Rank Vector, Zh. Vych. Mat. Mat. Fiz., 2015, vol. 55, no. 3, pp. 355–371.

    Google Scholar 

  18. Ermakov, S.M., Metod Monte-Karlo v vychislitel’noi matematike (The Monte Carlo Method in Computational Mathematics. Introductory Course), Moscow: Binom, 2009.

    Google Scholar 

  19. Zang, V.-B., Sinergeticheskaya ekonomika: vremya i peremeny v nelineinoi ekonomicheskoi teorii (Synergetic Economy: Time and Changes in Nonlinear Economic Theory), Moscow: Mir, 1999.

    Google Scholar 

  20. Zorich, V.A., Matematicheskii analiz zadach estestvoznaniya (Mathematical Analysis of the Problems of Natural Science), Moscow: MTSNMO, 2008.

    Google Scholar 

  21. Ibragimov, I.A. and Has’minskii, R.Z., Asimptoticheskaya teoriya otsenivaniya (Asymptotic Theory of Estimation), Moscow: Nauka, 1979.

    MATH  Google Scholar 

  22. Kalinkin, A.V., Markov Branching Processes with Interaction, Usp. Mat. Nauk, 2002, vol. 57, no. 2(344), pp. 23–84.

    Article  MathSciNet  MATH  Google Scholar 

  23. Kats, M., Veroyatnost’ i smezhnye voprosy v fizike (Probability and Related Questions in Physics), Moscow: Mir, 1965.

    Google Scholar 

  24. Kel’bert, M.Ya. and Suhov, Yu.M., Markovskie tsepi kak otpravnaya tochka teorii sluchainykh protsessov i ih prilozhenii (Markov Chains as a Starting Point of the Theory of Stochastic Processes and Their Applications), vol. 2, Moscow: MTSNMO, 2010.

    Google Scholar 

  25. Krasnosel’skii, M.A. and Kreyn, S.G., A Remark on Distribution of Errors in Solving a System of Linear Equations Using Iterative Process, Usp. Mat. Nauk, 1952, vol. 7, no. 4(50), pp. 157–161.

    Google Scholar 

  26. Krasnosel’skii, M.A., Lifshits, E.A., and Sobolev A.V., Pozitivnye lineinye sistemy. Metod polozhitel’nykh operatorov (Positive Linear Systems. Method of Positive Operators), Moscow: Nauka, 1985.

    MATH  Google Scholar 

  27. Lagutin, M.B., Naglyadnaya matematicheskaya statistika: uchebnoe posobie (Illustrative Mathematical Statistics: A Tutorial), 3rd ed., Moscow: Binom, 2013.

    Google Scholar 

  28. Malyshev, V.A. and Pirogov, S.A., Reversibility and Irreversibility in Stochastic Chemical Kinetics, Usp. Mat. Nauk, 2008, vol. 63, no. 1(379), pp. 4–36.

    MathSciNet  MATH  Google Scholar 

  29. Nikaido, H., Vypuklye struktury i matematicheskaya ekonomika (Convex Structures and Economic Theory), Moscow: Mir, 1972.

    Google Scholar 

  30. Polyak, B.T. and Tremba, A.A., Solving the Page Rank Problem for Large Matrices by Means of Regularization, Avtomat. Telemekh., 2012, vol. 11, pp. 144–166.

    MATH  Google Scholar 

  31. Purmal’, A.P., Slobodetskaya, E.M., and Travin, S.O., Kak prevrashchayutsya veshchestva (How Substances Transform), Moscow: Nauka, 1984.

    Google Scholar 

  32. Razzhevaikin, V.N., Analiz modelei dinamiki populyatsii. Uchebnoe posobie (Analysis of Models of Population Dynamics. Tutorial), Moscow: MIPT, 2010.

    Google Scholar 

  33. Raigorodskii, A.M., Modeli interneta (Internet Models), Dolgoprudnyi: Intellekt, 2013.

    Google Scholar 

  34. Sanov, I.N., On the Probability of Large Deviations of Random Variables, Mat. Sb., 1957, vol. 42(84), no. 1, pp. 11–44.

    MathSciNet  Google Scholar 

  35. Sinay, Ya.G., How Mathematicians Study Chaos, Mat. Prosveshch., 2001, vol. 5, pp. 32–46.

    Google Scholar 

  36. Tikhomirov, V.M., Rasskazy o maksimumakh i minimumakh (Stories about Maximums and Minimums), Moscow: Nauka, 1986.

    Google Scholar 

  37. Avrachenkov, K. and Lebedev, D., Page Rank of Scale-Free Growing Networks, Internet Math., 2006, vol. 3, no. 2, pp. 207–232.

    Article  MathSciNet  MATH  Google Scholar 

  38. Aldous, D. and Fill, J., Reversible Markov Chains and Random Walks on Graphs, Berkeley: University of California, 2002.

    Google Scholar 

  39. Ball, K., An Elementary Introduction to Modern Convex Geometry, Flav. Geom., 1997, vol. 31, pp. 1–58.

    MathSciNet  MATH  Google Scholar 

  40. Blum, A., Hopcroft, J., and Kannan, R., Foundations of Data Science; http://www.cs.cornell.edu/jeh/book.pdf.

  41. Bogolubsky, L., Dvurechensky, P., Gasnikov, A., Gusev, G., Nesterov, Yu., Raigorodskii, A., Tikhonov, A., and Zhukovskii, M., Learning Supervised Page Rank with Gradient-Based and Gradient-Free Optimization Methods, The Thirtieth Annual Conf. on Neural Information Processing Systems (NIPS 2016), Barcelona, 2016.

    Google Scholar 

  42. Boucheron, S., Lugosi, G., and Massart, P., Concentration Inequalities: A Nonasymptotic Theory of Independence, Oxford University Press, 2013.

    Book  MATH  Google Scholar 

  43. Brin, S. and Page, L., The Anatomy of a Large-Scale Hypertextual Web Search Engine, Comput. Network ISDN Syst., 1998, vol. 30, nos. 1–7, pp. 107–117.

    Article  Google Scholar 

  44. Diaconis, P., The Markov Chain Monte Carlo Revolution, Bull. (New Ser.) AMS, 2009, vol. 49, no. 2, pp. 179–205.

    MathSciNet  MATH  Google Scholar 

  45. Ethier, N.S. and Kurtz, T.G., Markov Processes, New York: Wiley, 1986.

    Book  MATH  Google Scholar 

  46. Chung, F., Laplacians and the Cheeger Inequality for Directed Graphs, Ann. Combinator., 2005, vol. 9, no. 1, pp. 1–19.

    Article  MathSciNet  MATH  Google Scholar 

  47. Franceschet, M., Page Rank: Standing on the Shoulders of Giant, Comm. ACM, 2011, vol. 54, no. 6, pp. 92–101.

    Article  Google Scholar 

  48. Grechnikov, E.A., The Degree Distribution and the Number of Edges between Nodes of Given Degrees in the Buckley–Osthus Model of a Random Web Graph, Internet Math., 2012, vol. 8, no. 3, pp. 257–287.

    Article  MathSciNet  MATH  Google Scholar 

  49. Jaynes, E.T., Probability Theory. The Logic of Science, Cambridge University Press, 2003.

    Book  MATH  Google Scholar 

  50. Joulin, A. and Ollivier, Y., Curvature, Concentration and Error Estimates for Markov Chain Monte Carlo, Ann. Probab., 2010, vol. 38, no. 6, pp. 2418–2442.

    Article  MathSciNet  MATH  Google Scholar 

  51. Kapur, J.N., Maximum—Entropy Models in Science and Engineering, New York: Wiley, 1989.

    MATH  Google Scholar 

  52. Langville, A.N. and Meyer, C.D., Google’s Page Rank and Beyond: The Science of Search Engine Rankings, Princeton University Press, 2006.

    MATH  Google Scholar 

  53. Ledoux, M., The Concentration of Measure Phenomenon, Math. Surveys Monogr., Providence, RI: AMS, 2001, vol. 89.

    Google Scholar 

  54. Levin, D.A., Peres, Y., and Wilmer, E.L., Markov Chain and Mixing Times, AMS, 2009.

    MATH  Google Scholar 

  55. Lezaud, P., Chernoff-Type Bound for Finite Markov Chains, Ann. Appl. Probab., 1998, vol. 8, no. 3, pp. 849–867.

    Article  MathSciNet  MATH  Google Scholar 

  56. Meyn, S.P. and Tweedie, R.L., Markov Chains and Stochastic Stability, London: Springer-Verlag, 2005.

    MATH  Google Scholar 

  57. Montenegro, R. and Tetali, P., Mathematical Aspects of Mixing Times in Markov Chains, Found. Trends Theor. Comp. Sci., 2016, vol. 1, no. 3, pp. 237–354.

    Article  MathSciNet  MATH  Google Scholar 

  58. Nesterov, Yu. and Nemirovski, A., Finding the Stationary States of Markov Chains by Iterative Methods, Appl. Math. Comput., 2015, vol. 255, pp. 58–65.

    MathSciNet  MATH  Google Scholar 

  59. Pandurangan, G., Raghavan, P., and Upfal, E., Using Page Rank to Characterize Web Structure, Internet Math., 2006, vol. 3, no. 1, pp. 1–20.

    Article  MathSciNet  MATH  Google Scholar 

  60. Paulin, D., Concentration Inequalities for Markov Chains by Marton Couplings, El. J. Probab., 2015, vol. 20, no. 79, pp. 1–32.

    MathSciNet  MATH  Google Scholar 

  61. Sandholm, W., Population Games and Evolutionary Dynamics. Economic Learning and Social Evolution, Cambridge: MIT Press, 2010.

    MATH  Google Scholar 

  62. Spielman, D., Lecture no. 7, 2009; http://www.cse.cuhk.edu.hk/chi/csc5160/notes/L07.pdf.

    Google Scholar 

  63. Spokoiny, V., Parametric Estimation. Finite Sample Theory, Ann. Stat., 2012, vol. 10, no. 6, pp. 2877–2909.

    Article  MathSciNet  MATH  Google Scholar 

  64. Van Handel, R., Probability in High Dimension, Princeton University, 2014.

    Book  Google Scholar 

  65. Chaos IX: Chaotic or not? Chaos research today; http://www.chaos-math.org/fr/chaos-ix-chaotique-oupas.

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Authors and Affiliations

  1. Moscow Institute of Physics and Technology, Institutskii per. 9, Dolgoprudnyi, Moscow Region, 141700, Russia

    A. V. Gasnikov, E. V. Gasnikova, A. A. M. Mohammed & E. O. Chernousova

  2. Institute for Information Transmission Problems, Russian Academy of Sciences, Bolshoi Karetnyi per. 19, build. 1, Moscow, 127051, Russia

    A. V. Gasnikov & P. E. Dvurechensky

  3. Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, Berlin, 10117, Germany

    P. E. Dvurechensky

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  1. A. V. Gasnikov
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  2. E. V. Gasnikova
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  3. P. E. Dvurechensky
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  4. A. A. M. Mohammed
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  5. E. O. Chernousova
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Gasnikov, A.V., Gasnikova, E.V., Dvurechensky, P.E. et al. About the Power Law of the PageRank Vector Component Distribution. Part 1. Numerical Methods for Finding the PageRank Vector. Numer. Analys. Appl. 10, 299–312 (2017). https://doi.org/10.1134/S1995423917040024

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