Generalisation of the Damping Factor in PageRank for Weighted Networks

  • Christopher EngströmEmail author
  • Sergei Silvestrov
Part of the EAA Series book series (EAAS)


In this article we will look at the PageRank algorithm used to rank nodes in a network. While the method was originally used by Brin and Page to rank home pages in order of “importance”, since then many similar methods have been used for other networks such as financial or P2P networks. We will work with a non-normalised version of the usual PageRank definition which we will then generalise to enable better options, such as adapting the method or allowing more types of data. We will show what kind of effects the new options creates using examples as well as giving some thoughts on what it can be used for. We will also take a brief look at how adding new connections between otherwise unconnected networks can change the ranking.



This research was supported in part by the Swedish Research Council (621- 2007-6338), the Swedish Foundation for International Cooperation in Research and Higher Education (STINT), the Royal Swedish Academy of Sciences, the Royal Physiographic Society in Lund and the Crafoord Foundation.


  1. 1.
    Andersson, F., Silvestrov, S.: The mathematics of internet search engines. Acta Appl. Math. 104, 211–242 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Battiston, S., Puliga, M., Kaushik, R., Tasca, P., Calderelli, G., DebtRank.: Too Central to Fail? Financial Networks, the FED and Systemic Risk. Macmillan, London (2012)Google Scholar
  3. 3.
    Berman, A., Plemmons, R.J.: Nonnegative Matrices in the Mathematical Sciences. Society for Industrial Applied Mathematics, Philadelphia (1994)Google Scholar
  4. 4.
    Bianchini, M., Gori, M., Scarselli, F.: Inside PageRank. ACM Trans. Internet Technol. 5(1), 92–128 (2005)Google Scholar
  5. 5.
    Brin, S., Page, L.: The anatomy of a large-scale hypertextual web search engine. Comput. Netw. ISDN Syst. 30(1–7), 107–117 (1998)CrossRefGoogle Scholar
  6. 6.
    Bryan, K., Leise, T.: The \(\$25,000,000,000\) eigenvector: the linear algebra behind Google. SIAM Rev. 48(3), 569–581 (2006)Google Scholar
  7. 7.
    Engström, C.: PageRank as a solution to a linear system, PageRank in changing systems and non-normalized versions of PageRank, Masters Thesis, Lund University, (2011) (LUTFMA-3220-2011)Google Scholar
  8. 8.
    Gantmacher, F.R.: The Theory of Matrices. Gantmacher, Chelsea (1959)zbMATHGoogle Scholar
  9. 9.
    Haveliwala, T., Kamvar, S.: The second eigenvalue of the Google matrix. Technical Report 2003-36, Stanford InfoLab, Stanford University (2003)Google Scholar
  10. 10.
    Ishii, H., Tempo, R., Bai, E.-W., Dabbene, F.: Distributed randomized PageRank computation based on web aggregation. In: Proceedings of the 48th IEEE Conference on Decision and Control Held Jointly with the 2009 28th Chinese Control Conference (CDC/CCC 2009), pp. 3026–3031 (2009)Google Scholar
  11. 11.
    James, R.N.: Markov Chains. Cambridge University Press, Cambridge (2009)Google Scholar
  12. 12.
    Kamvar, S., Haveliwala, T.: The condition number of the PageRank problem. Technical Report 2003-20, Stanford InfoLab (2003)Google Scholar
  13. 13.
    Kamvar, S., Haveliwala, T., Golub, G.: Adaptive methods for the computation of PageRank. Linear Algebra Appl. 386, 51–65 (2004)Google Scholar
  14. 14.
    Kamvar, S.D., Schlosser, M.T., Garcia-Molina, H.: The Eigentrust algorithm for reputation management in P2P networks. In: Proceedings of the 12th International Conference on World Wide Web, ACM, pp. 640–651. Budapest (2003)Google Scholar
  15. 15.
    Lancaster, P.: Theory of Matrices. Academic Press, New York (1969)Google Scholar
  16. 16.
    Page, L., Brin, S., Motwani, R., Winograd, T.: The PageRank citation ranking: bringing order to the web. Technical Report 1999–66, Stanford InfoLab (1999)Google Scholar
  17. 17.
    Rydén, T., Lindgren, G.: Markov Processer (in Swedish). Lund University, Lund (2000)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Mälardalen UniversityVästeråsSweden

Personalised recommendations