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Random Graphs Associated to Some Discrete and Continuous Time Preferential Attachment Models

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Abstract

We give a common description of Simon, Barabási–Albert, II-PA and Price growth models, by introducing suitable random graph processes with preferential attachment mechanisms. Through the II-PA model, we prove the conditions for which the asymptotic degree distribution of the Barabási–Albert model coincides with the asymptotic in-degree distribution of the Simon model. Furthermore, we show that when the number of vertices in the Simon model (with parameter \(\alpha \)) goes to infinity, a portion of them behave as a Yule model with parameters \((\lambda ,\beta ) = (1-\alpha ,1)\), and through this relation we explain why asymptotic properties of a random vertex in Simon model, coincide with the asymptotic properties of a random genus in Yule model. As a by-product of our analysis, we prove the explicit expression of the in-degree distribution for the II-PA model, given without proof in Newman (Contemp Phys 46:323-351, 2005). References to traditional and recent applications of the these models are also discussed.

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References

  1. Barabási, A., Albert, R.: Emergence of scaling in random networks. Science 286(5439), 509–512 (1999)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  2. Billingsley, P.: Probability and Measure, 3rd edn. Wiley, New York (1995)

    MATH  Google Scholar 

  3. Bollobás, B., Riordan, O., Spencer, J., Tusnády, G.: The degree sequence of a scale-free random graph process. Random Struct. Algorithms 18(3), 279–290 (2001)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  ADS  Google Scholar 

  5. Cooper, C., Frieze, A.: A general model of web graphs. Random Struct. Algorithms 22(3), 311–335 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  6. Crump, K.S.: On point processes having an order statistic structure. Sankhya Ser. A 37(3), 396–404 (1975)

    MathSciNet  MATH  Google Scholar 

  7. de Solla, D.J.: Price: Networks of scientific papers. Science 149, 510–515 (1965)

    Article  ADS  Google Scholar 

  8. de Solla, D.J.: Price: A general theory of bibliometric and other cumulative advantage processes. J. Am. Soc. Inform. Sci. 27, 292–306 (1976)

    Article  Google Scholar 

  9. Dorogovtsev, S.N., Mendes, J.F.F., Samukhin, A.N.: Structure of growing networks with preferential linking. Phys. Rev. Lett. 85, 4633–4636 (2000)

    Article  ADS  Google Scholar 

  10. Durrett, R.: Random Graph Dynamics. Cambridge University Press, New York (2006)

    Book  MATH  Google Scholar 

  11. Feigin, P.D.: On the characterization of point processes with the order statistic property. J. Appl. Probab. 16(2), 297–304 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  12. Janson, S., Łuczak, T., Ruciński, A.: Random Graphs. Wiley, New York (2011)

    MATH  Google Scholar 

  13. Lansky, P., Polito, F., Sacerdote, L.: The role of detachment of in-links in scale-free networks. J. Phys. A 47(34), 345002 (2014)

    Article  MATH  Google Scholar 

  14. Neuts, M., Resnick, S.I.: On the times of birth in a linear birth process. J. Aust. Math. Soc. 12, 473–475 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  15. Newman, M.E.J.: The structure and function of complex networks. SIAM Rev. 45, 167–256 (2003)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  16. Newman, M.E.J.: Power laws, Pareto distributions and Zipf’s law. Contemp. Phys. 46(5), 323–351 (2005)

    Article  ADS  Google Scholar 

  17. Ostroumova, L., Ryabchenko, A., Samosvat, E.: Generalized preferential attachment: tunable power-law degree distribution and clustering coefficient (2012). arXiv:1205.3015

  18. Puri, P.S.: On the characterization of point processes with the order statistic property without the moment condition Sankhya. J. Appl. Probab. 19(1), 39–51 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  19. Sedgewick, R.: An introduction to the analysis of algorithms. Addison-Wesley, Reading (2013)

    Google Scholar 

  20. Simkin, M.V., Roychowdhury, V.P.: Re-inventing willis. Phys. Rep. 502(1), 1–35 (2011)

    ADS  MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  22. Thompson, E.A.: Human Evolutionary Trees. Cambridge University Press, New York (1975)

    Google Scholar 

  23. van derHofstad, R.: Random Graphs and Complex Networks, Volume I. http://www.win.tue.nl/rhofstad/NotesRGCN.pdf (2014)

  24. Yule, G.U.: A mathematical theory of evolution, based on the conclusions of Dr. J. C. Willis, F.R.S. Philos. Trans. R. Soc. Lond. B 213(402–410), 21–87 (1925)

    Article  ADS  Google Scholar 

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Authors and Affiliations

  1. Mathematics Department “G. Peano”, University of Torino, Turin, Italy

    Angelica Pachon, Federico Polito & Laura Sacerdote

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  1. Angelica Pachon
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  2. Federico Polito
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  3. Laura Sacerdote
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Correspondence to Angelica Pachon.

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Pachon, A., Polito, F. & Sacerdote, L. Random Graphs Associated to Some Discrete and Continuous Time Preferential Attachment Models. J Stat Phys 162, 1608–1638 (2016). https://doi.org/10.1007/s10955-016-1462-7

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  • DOI: https://doi.org/10.1007/s10955-016-1462-7

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