Skip to main content

A Version of Herbert A. Simon’s Model with Slowly Fading Memory and Its Connections to Branching Processes

Abstract

Construct recursively a long string of words \(w_1,\ldots w_n,\) such that at each step k\(w_{k+1}\) is a new word with a fixed probability \(p\in (0,1),\) and repeats some preceding word with complementary probability \(1-p.\) More precisely, given a repetition occurs, \(w_{k+1}\) repeats the jth word with probability proportional to \(j^{\alpha }\) for \(j=1,\ldots , k.\) We show that the proportion of distinct words occurring exactly \(\ell \) times converges as the length n of the string goes to infinity to some probability mass function in the variable \(\ell \ge 1,\) whose tail decays as a power function when \(p<1/(1+\alpha ),\) and exponentially fast when \(p>1/(1+\alpha ).\)

This is a preview of subscription content, access via your institution.

References

  1. Athreya, K.B., Ney, P.E.: Branching processes. Die Grundlehren der mathematischen Wissenschaften, Band 196. Springer, New York (1972)

    Book  MATH  Google Scholar 

  2. Bornholdt, S., Ebel, H.: World Wide Web scaling exponent from Simon’s 1955 model. Phys. Rev. E 64, 035104 (2001)

    ADS  Article  Google Scholar 

  3. Cattuto, C., Loreto, V., Servedio, V.D.P.: A Yule–Simon process with memory. Europhys. Lett. 76(2), 208 (2006)

    ADS  MathSciNet  Article  Google Scholar 

  4. Dorogovtsev, S.N., Mendes, J.F.F.: Evolution of networks with aging of sites. Phys. Rev. E 62, 1842–1845 (2000)

    ADS  Article  Google Scholar 

  5. Dorogovtsev, S.N., Mendes, J.F.F.: Evolution of Networks: From Biological Nets to the Internet and WWW. Oxford University Press, Oxford (2003)

    Book  MATH  Google Scholar 

  6. Durrett, R.: Random Graph Dynamics. Cambridge Series in Statistical and Probabilistic Mathematics, vol. 20. Cambridge University Press, Cambridge (2007)

    MATH  Google Scholar 

  7. Garavaglia, A., van der Hofstad, R., Woeginger, G.: The dynamics of power laws: fitness and aging in preferential attachment trees. J. Stat. Phys. 168(6), 1137–1179 (2017)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  8. Jagers, P.: General branching processes as Markov fields. Stoch. Process. Appl. 32(2), 183–212 (1989)

    MathSciNet  Article  MATH  Google Scholar 

  9. Kallenberg, O.: Foundations of Modern Probability. Probability and Its Applications (New York), 2nd edn. Springer, New York (2002)

    Book  MATH  Google Scholar 

  10. Kyprianou, A.E.: Fluctuations of Lévy Processes with Applications. Introductory Lectures. Universitext, 2nd edn. Springer, Heidelberg (2014)

    Book  MATH  Google Scholar 

  11. Lansky, P., Polito, F., Sacerdote, L.: Generalized nonlinear Yule models. J. Stat. Phys. 165(3), 661–679 (2016)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  12. Nerman, O.: On the convergence of supercritical general (C–M–J) branching processes. Z. Wahrsch. Verwandte Geb. 57(3), 365–395 (1981)

    MathSciNet  Article  MATH  Google Scholar 

  13. Pachon, A., Polito, F., Sacerdote, L.: Random graphs associated to some discrete and continuous time preferential attachment models. J. Stat. Phys. 162(6), 1608–1638 (2016)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  14. Polito, F.: Studies on generalized Yule models. Mod. Stoch. Theory Appl. 6(1), 1–55 (2019)

    MathSciNet  Google Scholar 

  15. Schaigorodsky, A.L., Perotti, J.I., Almeira, N., Billoni, O.V.: Short-ranged memory model with preferential growth. Phys. Rev. E 97, 022132 (2018)

    ADS  Article  Google Scholar 

  16. Simon, H.A.: On a class of skew distribution functions. Biometrika 42(3/4), 425–440 (1955)

    MathSciNet  Article  MATH  Google Scholar 

  17. Sun, J., Staab, S., Karimi, F.: Decay of relevance in exponentially growing networks. In: Proceedings of the 10th ACM Conference on Web Science, WebSci ’18, New York, NY, USA, pp. 343–351. ACM (2018)

  18. van der Hofstad, R.: Random Graphs and Complex Networks. Cambridge Series in Statistical and Probabilistic Mathematics, [43], vol. 1. Cambridge University Press, Cambridge (2017)

    Book  MATH  Google Scholar 

  19. Wang, M., Yu, G., Yu, D.: Measuring the preferential attachment mechanism in citation networks. Physica A 387(18), 4692–4698 (2008)

    ADS  Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Jean Bertoin.

Additional information

Communicated by Eric A. A. Carlen.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Bertoin, J. A Version of Herbert A. Simon’s Model with Slowly Fading Memory and Its Connections to Branching Processes. J Stat Phys 176, 679–691 (2019). https://doi.org/10.1007/s10955-019-02316-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10955-019-02316-1

Keywords

  • Yule–Simon model
  • Preferential attachment
  • Memory
  • Continuous state branching process
  • Crump–Mode–Jagers branching process
  • Heavy tail distributions

Mathematics Subject Classification

  • 60J85
  • 05C85