PageRank in Scale-Free Random Graphs

  • Ningyuan Chen
  • Nelly Litvak
  • Mariana Olvera-CraviotoEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8882)


We analyze the distribution of PageRank on a directed configuration model and show that as the size of the graph grows to infinity, the PageRank of a randomly chosen node can be closely approximated by the PageRank of the root node of an appropriately constructed tree. This tree approximation is in turn related to the solution of a linear stochastic fixed-point equation that has been thoroughly studied in the recent literature.


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  1. 1.
    Alsmeyer, G., Damek, E., Mentemeier, S.: Tails of fixed points of the two-sided smoothing transform. In: Springer Proceedings in Mathematics & Statistics: Random Matrices and Iterated Random Functions (2012)Google Scholar
  2. 2.
    Alsmeyer, G., Meiners, M.: Fixed points of the smoothing transform: Two-sided solutions. Probab. Theory Relat. Fields 155(1–2), 165–199 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Avrachenkov, K., Lebedev, D.: PageRank of scale-free growing networks. Internet Mathematics 3(2), 207–231 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Brin, S., Page, L.: The anatomy of a large-scale hypertextual Web search engine. Computer Networks and ISDN Systems 33, 107–117 (1998)CrossRefGoogle Scholar
  5. 5.
    Chen, N., Litvak, N., Olvera-Cravioto, M.: Ranking algorithms on directed configuration networks. ArXiv:1409.7443, pp. 1–39 (2014)
  6. 6.
    Chen, N., Olvera-Cravioto, M.: Directed random graphs with given degree distributions. Stochastic Systems 3(1), 147–186 (2013)CrossRefzbMATHGoogle Scholar
  7. 7.
    Jelenković, P.R., Olvera-Cravioto, M.: Information ranking and power laws on trees. Adv. Appl. Prob. 42(4), 1057–1093 (2010)CrossRefzbMATHGoogle Scholar
  8. 8.
    Jelenković, P.R., Olvera-Cravioto, M.: Implicit renewal theory and power tails on trees. Adv. Appl. Prob. 44(2), 528–561 (2012)CrossRefzbMATHGoogle Scholar
  9. 9.
    Jelenković, P.R., Olvera-Cravioto, M.: Implicit renewal theory for trees with general weights. Stochastic Process. Appl. 122(9), 3209–3238 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Langville, A.N., Meyer, C.D.: Google PageRank and beyond. Princeton University Press (2006)Google Scholar
  11. 11.
    Litvak, N., Scheinhardt, W.R.W., Volkovich, Y.: In-degree and PageRank: Why do they follow similar power laws? Internet Mathematics 4(2), 175–198 (2007)CrossRefMathSciNetGoogle Scholar
  12. 12.
    Olvera-Cravioto, M.: Tail behavior of solutions of linear recursions on trees. Stochastic Process. Appl. 122(4), 1777–1807 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Pandurangan, G., Raghavan, P., Upfal, E.: Using PageRank to characterize web structure. In: Ibarra, O.H., Zhang, L. (eds.) COCOON 2002. LNCS, vol. 2387, pp. 330–339. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  14. 14.
    van der Hofstad, R.: Random graphs and complex networks (2009)Google Scholar
  15. 15.
    Volkovich, Y., Litvak, N.: Asymptotic analysis for personalized web search. Adv. Appl. Prob. 42(2), 577–604 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Volkovich, Y., Litvak, N., Donato, D.: Determining factors behind the pagerank log-log plot. In: Proceedings of the 5th International Workshop on Algorithms and Models for the Web-graph, pp. 108–123 (2007)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Ningyuan Chen
    • 1
  • Nelly Litvak
    • 2
  • Mariana Olvera-Cravioto
    • 1
    Email author
  1. 1.Columbia UniversityNew YorkUSA
  2. 2.University of TwenteEnschedeThe Netherlands

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