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Preferential Attachment Random Graphs with Edge-Step Functions

  • Caio Alves
  • Rodrigo RibeiroEmail author
  • Rémy Sanchis
Article
  • 4 Downloads

Abstract

We analyze a random graph model with preferential attachment rule and edge-step functions that govern the growth rate of the vertex set, and study the effect of these functions on the empirical degree distribution of these random graphs. More specifically, we prove that when the edge-step function f is a monotone regularly varying function at infinity, the degree sequence of graphs associated with it obeys a (generalized) power-law distribution whose exponent belongs to (1, 2] and is related to the index of regular variation of f at infinity whenever said index is greater than \(-1\). When the regular variation index is less than or equal to \(-1\), we show that the empirical degree distribution vanishes for any fixed degree.

Keywords

Complex networks Preferential attachment Concentration bounds Power-law Scale-free Karamata’s theory Regularly varying functions 

Mathematics Subject Classification (2010)

Primary 05C82 Secondary 60K40 68R10 

Notes

Acknowledgements

C. A. was supported by the Deutsche Forschungsgemeinschaft (DFG). R. R. was partially supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq). R.S. has been partially supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) and by FAPEMIG (Programa Pesquisador Mineiro), Grant PPM 00600/16.

References

  1. 1.
    Alves, C., Ribeiro, R., Sanchis, R.: Large communities in a scale-free network. J. Stat. Phys. 166(1), 137–149 (2017)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Alves, C., Ribeiro, R., Sanchis, R.: Agglomeration in a preferential attachment random graph with edge-steps. (2019). Preprint. arxiv:1901.02486Google Scholar
  3. 3.
    Alves, C., Ribeiro, R., Sanchis, R.: Topological properties of p.a. random graphs with edge-step functions (2019). Preprint. arxiv:1902.10165Google Scholar
  4. 4.
    Barabási, A.-L., Albert, R.: Emergence of scaling in random networks. Science (1999)Google Scholar
  5. 5.
    Bingham, N.H., Goldie, C.M., Teugels, J.L.: Regular Variation. Encyclopedia of Mathematics and its Applications, vol. 27. CUP, Cambridge (1989)zbMATHGoogle Scholar
  6. 6.
    Bollobás, B., Riordan, O., Spencer, J., Tusnády, G.: The degree sequence of a scale-free random graph process. Random Struct. Algorithms 18, 279–290 (2001)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Chung, F., Lu, L.: Complex Graphs and Networks (Cbms Regional Conference Series in Mathematics). American Mathematical Society, Boston (2006)Google Scholar
  8. 8.
    Cooper, C., Frieze, A.: A general model of web graphs. Random Struct. Algorithms 22(3), 311–335 (2003)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Crane, Harry, Dempsey, Walter: Edge exchangeable models for interaction networks. J. Am. Stat. Assoc. 113(523), 1311–1326 (2018)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Deijfen, M., van den Esker, H., van der Hofstad, R., Hooghiemstra, G.: A preferential attachment model with random initial degrees. Arkiv för Matematik 47(1), 41–72 (2009)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Dereich, S., Ortgiese, M.: Robust analysis of preferential attachment models with fitness. Comb. Probab. Comput. 23(3), 386–411 (2014)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Freedman, D.A.: On tail probabilities for martingales. Ann. Probab. 3(1), 100–118 (1975)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Jacob, E., Mörters, P.: Spatial preferential attachment networks: power laws and clustering coefficients. Ann. Appl. Probab. 25(2), 632–662, 04 (2015)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Kim, B., Holme, P.: Growing scale-free networks with tunable clustering. Phys. Rev. E (2002)Google Scholar
  15. 15.
    Malyshkin, Y., Paquette, E.: The power of choice combined with preferential attachement. Electron. Commun. Probab. 19(44), 1–13 (2014)zbMATHGoogle Scholar
  16. 16.
    Pew Research Center. Social media update 2014 (2014)Google Scholar
  17. 17.
    Strogatz, S.H., Watts, D.J.: Tcollective dynamics of ’small-world’ networks. Nature (1998)Google Scholar
  18. 18.
    Thörnblad, E.: Asymptotic degree distribution of a duplication-deletion random graph model. Int. Math. 11, 03 (2014)MathSciNetGoogle Scholar
  19. 19.
    Van Der Hofstad, R.: Random graphs and complex networks. Available on http://www.win.tue.nl/rhofstad/NotesRGCN.pdf (2009)
  20. 20.
    Wang, W.-Q., Zhang, Q.-M., Zhou, T.: Evaluating network models: a likelihood analysis. EPL (Europhysics Letters) 98(2), 28004 (2012)CrossRefGoogle Scholar

Copyright information

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

Authors and Affiliations

  1. 1.Institute of MathematicsUniversity of LeipzigLeipzigGermany
  2. 2.IMPA, Estrada Da. CastorinaRio de JaneiroBrazil
  3. 3.Departamento de MatemáticaUniversidade Federal de Minas GeraisBelo HorizonteBrazil

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