Skip to main content
SpringerLink
Log in

Large-deviation properties of the largest biconnected component for random graphs

  • Regular Article
  • Published:
The European Physical Journal B Aims and scope Submit manuscript

Abstract

We study the size of the largest biconnected components in sparse Erdős–Rényi graphs with finite connectivity and Barabási–Albert graphs with non-integer mean degree. Using a statistical-mechanics inspired Monte Carlo approach we obtain numerically the distributions for different sets of parameters over almost their whole support, especially down to the rare-event tails with probabilities far less than 10−100. This enables us to observe a qualitative difference in the behavior of the size of the largest biconnected component and the largest 2-core in the region of very small components, which is unreachable using simple sampling methods. Also, we observe a convergence to a rate function even for small sizes, which is a hint that the large deviation principle holds for these distributions.

Graphical abstract

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. M.E.J. Newman, SIAM Rev. 45, 167 (2003)

    Article  ADS  MathSciNet  Google Scholar 

  2. S.N. Dorogovtsev, J.F.F. Mendes, Evolution of Networks: From Biological Nets to the Internet and WWW (Oxford University Press, Oxford, 2006)

  3. M. Newman, A.L. Barabási, D. Watts, The Structure and Dynamics of Networks (Princeton University Press, 2006)

  4. M. Newman, Networks: an Introduction (Oxford University Press, Princeton, 2010)

  5. A. Barrat, M. Barthélemy, A. Vespignani, Dynamical Processes on Complex Networks (Cambridge University Press, Cambridge, 2012)

  6. M.L. Sachtjen, B.A. Carreras, V.E. Lynch, Phys. Rev. E 61, 4877 (2000)

    Article  ADS  Google Scholar 

  7. M. Rohden, A. Sorge, M. Timme, D. Witthaut, Phys. Rev. Lett. 109, 064101 (2012)

    Article  ADS  Google Scholar 

  8. T. Dewenter, A.K. Hartmann, New J. Phys. 17, 015005 (2015)

    Article  ADS  Google Scholar 

  9. R. Cohen, K. Erez, D. ben Avraham, S. Havlin, Phys. Rev. Lett. 85, 4626 (2000)

    Article  ADS  Google Scholar 

  10. D.S. Lee, H. Rieger, EPL (Europhys. Lett.) 73, 471 (2006)

    Article  ADS  Google Scholar 

  11. C.M. Ghim, K.I. Goh, B. Kahng, J. Theor. Biol. 237, 401 (2005)

    Article  Google Scholar 

  12. P. Kim, D.S. Lee, B. Kahng, Sci. Rep. 5, 15567 (2015)

    Article  ADS  Google Scholar 

  13. P. Erdős, A. Rényi, Publ. Math. Inst. Hungar. Acad. Sci. 5, 17 (1960)

    Google Scholar 

  14. D.J. Watts, S.H. Strogatz, Nature 393, 440 (1998)

    Article  ADS  Google Scholar 

  15. A.L. Barabási, R. Albert, Science 286, 509 (1999)

    Article  ADS  MathSciNet  Google Scholar 

  16. R. Albert, H. Jeong, A.L. Barabási, Nature 406, 378 (2000)

    Article  ADS  Google Scholar 

  17. D.S. Callaway, M.E.J. Newman, S.H. Strogatz, D.J. Watts, Phys. Rev. Lett. 85, 5468 (2000)

    Article  ADS  Google Scholar 

  18. M.E.J. Newman, G. Ghoshal, Phys. Rev. Lett. 100, 138701 (2008)

    Article  ADS  Google Scholar 

  19. C. Norrenbrock, O. Melchert, A.K. Hartmann, Phys. Rev. E 94, 062125 (2016)

    Article  ADS  Google Scholar 

  20. G. Bianconi, Phys. Rev. E 97, 022314 (2018)

    Article  ADS  Google Scholar 

  21. G. Bianconi, Phys. Rev. E 96, 012302 (2017)

    Article  ADS  Google Scholar 

  22. P. Kim, D.S. Lee, B. Kahng, Phys. Rev. E 87, 022804 (2013)

    Article  ADS  Google Scholar 

  23. M. Biskup, L. Chayes, S.A. Smith, Random Struct. Algor. 31, 354 (2007)

    Article  Google Scholar 

  24. A.K. Hartmann, Eur. Phys. J. B 84, 627 (2011)

    Article  ADS  MathSciNet  Google Scholar 

  25. A.K. Hartmann, Eur. Phys. J. Special Topics 226, 567 (2017)

    Article  ADS  Google Scholar 

  26. A.K. Hartmann, Big Practical Guide to Computer Simulations (World Scientific, Singapore, 2015)

  27. F. den Hollander Large Deviations (American Mathematical Society, Providence, 2000)

  28. H. Touchette, Phys. Rep. 478, 1 (2009)

    Article  ADS  MathSciNet  Google Scholar 

  29. J. Hopcroft, R. Tarjan, Commun. ACM 16, 372 (1973)

    Article  Google Scholar 

  30. T.H. Cormen, C.E. Leiserson, R.L. Rivest, C. Stein, Introduction to Algorithms (MIT Press, MA, 2009)

  31. B. Dezső, A. Jüttner, P. Kovács, Electron. Notes Theor. Comput. Sci. 264, 23 (2011) (Proceedings of the Second Workshop on Generative Technologies (WGT) 2010)

    Article  Google Scholar 

  32. G. Ghoshal, Ph.D. thesis, University of Michigan, 2009

  33. M.E.J. Newman, S.H. Strogatz, D.J. Watts, Phys. Rev. E 64, 026118 (2001)

    Article  ADS  Google Scholar 

  34. H. Klein-Hennig, A.K. Hartmann, Phys. Rev. E 85, 026101 (2012)

    Article  ADS  Google Scholar 

  35. A.K. Hartmann, Phys. Rev. E 65, 056102 (2002)

    Article  ADS  Google Scholar 

  36. S. Wolfsheimer, B. Burghardt, A.K. Hartmann, Algorithm. Mol. Biol. 2, 9 (2007)

    Article  Google Scholar 

  37. P. Fieth, A.K. Hartmann, Phys. Rev. E 94, 022127 (2016)

    Article  ADS  Google Scholar 

  38. G. Claussen, A.K. Hartmann, S.N. Majumdar, Phys. Rev. E 91, 052104 (2015)

    Article  ADS  MathSciNet  Google Scholar 

  39. T. Dewenter, G. Claussen, A.K. Hartmann, S.N. Majumdar, Phys. Rev. E 94, 052120 (2016)

    Article  ADS  MathSciNet  Google Scholar 

  40. H. Schawe, A.K. Hartmann, S.N. Majumdar, Phys. Rev. E 97, 062159 (2018)

    Article  ADS  Google Scholar 

  41. H. Schawe, A.K. Hartmann, https://doi.org/arXiv:1808.10698 (2018)

  42. A.K. Hartmann, Phys. Rev. E 89, 052103 (2014)

    Article  ADS  Google Scholar 

  43. A. Engel, R. Monasson, A.K. Hartmann, J. Stat. Phys. 117, 387 (2004)

    Article  ADS  Google Scholar 

  44. A.K. Hartmann, M. Mézard, Phys. Rev. E 97, 032128 (2018)

    Article  ADS  Google Scholar 

  45. F. Wang, D.P. Landau, Phys. Rev. Lett. 86, 2050 (2001)

    Article  ADS  Google Scholar 

  46. F. Wang, D.P. Landau, Phys. Rev. E 64, 056101 (2001)

    Article  ADS  Google Scholar 

  47. B.J. Schulz, K. Binder, M. Müller, D.P. Landau, Phys. Rev. E 67, 067102 (2003)

    Article  ADS  Google Scholar 

  48. R.E. Belardinelli, V.D. Pereyra, Phys. Rev. E 75, 046701 (2007)

    Article  ADS  Google Scholar 

  49. R.E. Belardinelli, V.D. Pereyra, J. Chem. Phys. 127, 184105 (2007)

    Article  ADS  Google Scholar 

  50. J. Lee, Phys. Rev. Lett. 71, 211 (1993)

    Article  ADS  Google Scholar 

  51. R. Dickman, A.G. Cunha-Netto, Phys. Rev. E 84, 026701 (2011)

    Article  ADS  Google Scholar 

  52. B. Efron, Ann. Stat. 7, 1 (1979)

    Article  Google Scholar 

  53. A.P. Young, Everything You Wanted to Know About Data Analysis and Fitting but Were Afraid to Ask, SpringerBriefs in Physics (Springer International Publishing, Switzerland, 2015)

Download references

Author information

Authors and Affiliations

  1. Institut für Physik, Universität Oldenburg, 26111, Oldenburg, Germany

    Hendrik Schawe & Alexander K. Hartmann

Authors
  1. Hendrik Schawe
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Alexander K. Hartmann
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hendrik Schawe.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Schawe, H., Hartmann, A.K. Large-deviation properties of the largest biconnected component for random graphs. Eur. Phys. J. B 92, 73 (2019). https://doi.org/10.1140/epjb/e2019-90667-y

Download citation

  • Received:

  • Revised:

  • Published:

  • DOI: https://doi.org/10.1140/epjb/e2019-90667-y

Keywords

Search

Navigation