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Acta Mathematica Hungarica

, Volume 134, Issue 1–2, pp 45–53 | Cite as

When the degree sequence is a sufficient statistic

  • Villő Csiszár
  • Péter Hussami
  • János Komlós
  • Tamás F. Móri
  • Lídia RejtőEmail author
  • Gábor Tusnády
Article

Abstract

There is a uniquely defined random graph model with independent adjacencies in which the degree sequence is a sufficient statistic. The model was recently discovered independently by several authors. Here we join to the statistical investigation of the model, proving that if the degree sequence is in the interior of the polytope defined by the Erdős–Gallai conditions, then a unique maximum likelihood estimate exists.

Keywords

degree sequence of graphs random graph sufficient statistics maximum likelihood estimation 

2000 Mathematics Subject Classification

62F10 05C07 

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References

  1. [1]
    A. Barvinok and J. A. Hartigan, An asymptotic formula for the number of non-negative integer matrices with prescribed row and column sums, preprint (2009), http://arxiv.org/abs/0910.2477.
  2. [2]
    A. Barvinok and J. A. Hartigan, The number of graphs and a random graph with a given degree sequence, preprint (2010), http://arxiv.org/abs/1003.0356.
  3. [3]
    S. Chatterjee, P. Diaconis and A. Sly, Random graphs with a given degree sequence, preprint (2010, arXiv: 1005.1136v3 [math.PR]).
  4. [4]
    I. Csiszár and P. Shields, Information theory and statistics: A tutorial, Foundations and Trends in Communications and Information Theory, 1(4) (2004), 417–528. now Publishers CrossRefGoogle Scholar
  5. [5]
    V. Csiszár, Conditional independence relations and log-linear models for random matchings, Acta Math. Hungar., 122 (2009), 131–152. CrossRefzbMATHMathSciNetGoogle Scholar
  6. [6]
    V. Csiszár, L. Rejtő and G. Tusnády, Statistical inference on random structures, in: E. Győri et al. (Eds.), Horizon of Combinatorics (2008), pp. 37–67. CrossRefGoogle Scholar
  7. [7]
    P. Erdős and T. Gallai, Graphs with given degrees of vertices, Mat. Lapok, 11 (1960), 264–274 (in Hungarian). Google Scholar
  8. [8]
    P. Hussami, Statistical inference on random graphs, PhD Thesis, 2010 (submitted to Central European University, Budapest). Google Scholar
  9. [9]
    M. Koren, Extreme degree sequences of simple graphs, J. Combinatorial Theory B, 15 (1973), 213–224. CrossRefzbMATHMathSciNetGoogle Scholar
  10. [10]
    M. Newman, A.-L. Barabási and D. Watts, The Structure and Dynamics of Networks, Princeton Studies in Complexity, Princeton University Press (2007). Google Scholar
  11. [11]
    G. Sierskma and H. Hoogeveen, Seven criteria for integer sequences being graphic, J. Graph Theory, 2 (1991), 223–231. Google Scholar

Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2011

Authors and Affiliations

  • Villő Csiszár
    • 1
  • Péter Hussami
    • 2
  • János Komlós
    • 3
  • Tamás F. Móri
    • 1
  • Lídia Rejtő
    • 2
    • 4
    Email author
  • Gábor Tusnády
    • 2
  1. 1.Eötvös Loránd UniversityBudapestHungary
  2. 2.Alfréd Rényi Institute of MathematicsHungarian Academy of SciencesBudapestHungary
  3. 3.Department of MathematicsRutgers UniversityNew BrunswickUSA
  4. 4.Statistics Program, FREC, CANRUniversity of DelawareNewarkUSA

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