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Large-deviation properties of the largest 2-core component for random graphs

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Abstract

Distributions of the size of the largest component of the 2-core for Erdos-Rényi (ER) random graphs with finite connectivity c and a finite number N of nodes are studied. The distributions are obtained basically over the full range of the support, with probabilities down to values as small as 10−320. This is achieved by using an artificial finite-temperature (Boltzmann) ensemble. The distributions for the 2-core resemble roughly the results obtained previously for the largest components of the full ER random graphs, but they are shifted to much smaller probabilities (c ≤ 1) or to smaller sizes (c > 1). The numerical data is compatible with a convergence of the rate function to a limiting shape, i.e., the large-deviations principle apparently holds.

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References

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

  2. D.C. Rapaport, The Art of Molecular Dynamics Simulation (Cambridge University Press, Cambridge, GB, 2004)

  3. J.N. Reddy, Introduction to the Finite Element Method (Mcgraw-Hill Education, Columbus, USA, 2005)

  4. M.E.J. Newman, G.T. Barkema, Monte Carlo Methods in Statistical Physics (Clarendon Press, Oxford, 1999)

  5. D.P. Landau, K. Binder, Monte Carlo Simulations in Statistical Physics (Cambridge University Press, Cambridge, 2000)

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

  7. A. Dembo, O. Zeitouni, Large Deviations Techniques and Applications (Springer, Berlin, 2010)

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

    Article  ADS  Google Scholar 

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

    Article  Google Scholar 

  10. L. Newberg, J. Comp. Biol. 15, 1187 (2008)

    Article  MathSciNet  Google Scholar 

  11. R. Durbin, S.R. Eddy, A. Krogh, G. Mitchison, Biological Sequence Analysis (Cambridge University Press, Cambridge, 2006)

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

    Article  ADS  Google Scholar 

  13. A.K. Hartmann, Phys. Rev. Lett. 94, 050601 (2005)

    Article  ADS  Google Scholar 

  14. M. Körner, H.G. Katzgraber, A.K. Hartmann, JSTAT 2006, P04005 (2006)

    Article  Google Scholar 

  15. C. Monthus, T. Garel, Phys. Rev. E 74, 051109 (2006)

    Article  ADS  Google Scholar 

  16. Y. Matsuda, H. Nishimori, K. Hukushima, J. Phys. A: Math. Theor. 41, 324012 (2008)

    Article  Google Scholar 

  17. Y. Iba, K. Hukushima, J. Phys. Soc. Japan 77, 103801 (2008)

    Article  ADS  Google Scholar 

  18. S. Wolfsheimer, A.K. Hartmann, Phys. Rev. E 82, 021902 (2010)

    Article  ADS  Google Scholar 

  19. T.A. Driscoll, K.L. Maki, SIAM Review 49, 673 (2007)

    Article  ADS  MathSciNet  Google Scholar 

  20. N. Saito, Y. Iba, K. Hukushima, Phys. Rev. E 82, 031142 (2010)

    Article  ADS  Google Scholar 

  21. A.K. Hartmann, S.N. Majumdar, A. Rosso, Phys. Rev. E 88, 022119 (2013)

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  24. J. Chalupa, P.L. Leath, G.R. Reich, J. Phys. C 12, L31 (1979) http://stacks.iop.org/0022-3719/12/i=1/a=008

    Article  ADS  Google Scholar 

  25. S.B. Seidman, Social Netw. 5, 269 (1983)

    Article  Google Scholar 

  26. G.D. Bader, C.W. Hogue, BMC Bioinformatics 4, 2 (2003)

    Article  Google Scholar 

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

    Google Scholar 

  28. N. O'Connell, Probab. Theo. Relat. Fields 110, 277 (1998)

    Article  Google Scholar 

  29. N. Metropolis, A.W. Rosenbluth, M.N. Rosenbluth, A. Teller, E. Teller, J. Chem. Phys. 21, 1087 (1953)

    Article  ADS  Google Scholar 

  30. A.M. Ferrenberg, R.H. Swendsen, Phys. Rev. Lett. 63, 1195 (1989)

    Article  ADS  Google Scholar 

  31. A.K. Hartmann, in New Optimization Algorithms in Physics, edited by A.K. Hartmann, H. Rieger (Wiley-VCH, Weinheim, 2004), p. 253

  32. B.A. Berg, T. Neuhaus, Phys. Rev. Lett. 68, 9 (1992)

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  34. B. Pittel, J. Spencer, N.C. Wormald, J. Comb. Theory B 67, 111 (1996)

    Article  Google Scholar 

  35. A.K. Hartmann, M. Weigt, Phase Transitions in Combinatorial Optimization Problems (Wiley-VCH, Weinheim, 2005)

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Authors and Affiliations

  1. Institut für Physik, Carl von Ossietzky Universität Oldenburg, 26111, Oldenburg, Germany

    Alexander K. Hartmann

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  1. Alexander K. Hartmann
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Correspondence to Alexander K. Hartmann.

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Hartmann, A.K. Large-deviation properties of the largest 2-core component for random graphs. Eur. Phys. J. Spec. Top. 226, 567–579 (2017). https://doi.org/10.1140/epjst/e2016-60368-3

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  • DOI: https://doi.org/10.1140/epjst/e2016-60368-3

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