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

  • Jean BertoinEmail author
Article
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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 \(p>1/(1+\alpha ).\)

Keywords

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

Mathematics Subject Classification

60J85 05C85 

Notes

References

  1. 1.
    Athreya, K.B., Ney, P.E.: Branching processes. Die Grundlehren der mathematischen Wissenschaften, Band 196. Springer, New York (1972)zbMATHGoogle Scholar
  2. 2.
    Bornholdt, S., Ebel, H.: World Wide Web scaling exponent from Simon’s 1955 model. Phys. Rev. E 64, 035104 (2001)ADSGoogle Scholar
  3. 3.
    Cattuto, C., Loreto, V., Servedio, V.D.P.: A Yule–Simon process with memory. Europhys. Lett. 76(2), 208 (2006)ADSMathSciNetGoogle Scholar
  4. 4.
    Dorogovtsev, S.N., Mendes, J.F.F.: Evolution of networks with aging of sites. Phys. Rev. E 62, 1842–1845 (2000)ADSGoogle Scholar
  5. 5.
    Dorogovtsev, S.N., Mendes, J.F.F.: Evolution of Networks: From Biological Nets to the Internet and WWW. Oxford University Press, Oxford (2003)zbMATHGoogle Scholar
  6. 6.
    Durrett, R.: Random Graph Dynamics. Cambridge Series in Statistical and Probabilistic Mathematics, vol. 20. Cambridge University Press, Cambridge (2007)zbMATHGoogle Scholar
  7. 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)ADSMathSciNetzbMATHGoogle Scholar
  8. 8.
    Jagers, P.: General branching processes as Markov fields. Stoch. Process. Appl. 32(2), 183–212 (1989)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Kallenberg, O.: Foundations of Modern Probability. Probability and Its Applications (New York), 2nd edn. Springer, New York (2002)zbMATHGoogle Scholar
  10. 10.
    Kyprianou, A.E.: Fluctuations of Lévy Processes with Applications. Introductory Lectures. Universitext, 2nd edn. Springer, Heidelberg (2014)zbMATHGoogle Scholar
  11. 11.
    Lansky, P., Polito, F., Sacerdote, L.: Generalized nonlinear Yule models. J. Stat. Phys. 165(3), 661–679 (2016)ADSMathSciNetzbMATHGoogle Scholar
  12. 12.
    Nerman, O.: On the convergence of supercritical general (C–M–J) branching processes. Z. Wahrsch. Verwandte Geb. 57(3), 365–395 (1981)MathSciNetzbMATHGoogle Scholar
  13. 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)ADSMathSciNetzbMATHGoogle Scholar
  14. 14.
    Polito, F.: Studies on generalized Yule models. Mod. Stoch. Theory Appl. 6(1), 1–55 (2019)MathSciNetGoogle Scholar
  15. 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)ADSGoogle Scholar
  16. 16.
    Simon, H.A.: On a class of skew distribution functions. Biometrika 42(3/4), 425–440 (1955)MathSciNetzbMATHGoogle Scholar
  17. 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)Google Scholar
  18. 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)zbMATHGoogle Scholar
  19. 19.
    Wang, M., Yu, G., Yu, D.: Measuring the preferential attachment mechanism in citation networks. Physica A 387(18), 4692–4698 (2008)ADSGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Institut für MathematikUniversität ZürichZurichSwitzerland

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