Skip to main content
SpringerLink
Log in

Degree-based moment estimation for ordered networks

  • Published:
Journal of Systems Science and Complexity Aims and scope Submit manuscript

Abstract

The edges between vertices in networks take not only the common binary values, but also the ordered values in some situations (e.g., the measurement of the relationship between people from worst to best in social networks). In this paper, the authors study the asymptotic property of the moment estimator based on the degrees of vertices in ordered networks whose edges are ordered random variables. In particular, the authors establish the uniform consistency and the asymptotic normality of the moment estimator when the number of parameters goes to infinity. Simulations and a real data example are provided to illustrate asymptotic results.

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. Watts D J, Small Worlds: The Dynamics of Networks between Order and Randomness, Princeton University Press, Princeton, 1999.

    MATH  Google Scholar 

  2. Christakis N A and Fowler J H, Connected: The Surprising Power of Our Social Networks and Now They Shape Our Lives, Little, Brown, 2009.

    Google Scholar 

  3. Kossinets G and Watts D J, Empirical analysis of an evolving social network, Science, 2006, 311(5757): 88–90.

    Article  MathSciNet  MATH  Google Scholar 

  4. Von Mering C, Krause R, and Snel B, Comparative assessment of large-scale data sets of proteinprotein interactions, Nature, 2002, 417(6887): 399–403.

    Article  Google Scholar 

  5. Bader G D and Hogue C W, An automated method for finding molecular complexes in large protein interaction networks, BMC Bioinformatics, 2003, 4(1): 2.

    Article  Google Scholar 

  6. Nepusz T, Yu H, and Paccanaro A, Detecting overlapping protein complexes in protein-protein interaction networks, Nature Methods, 2012, 9(5): 471–472.

    Article  Google Scholar 

  7. Yook S H, Oltvai Z N, and Barabsi A L, Functional and topological characterization of protein interaction networks, Proteomics, 2004, 4(4): 928–942.

    Article  Google Scholar 

  8. Blitzstein J and Diaconis P, A sequential importance sampling algorithm for generating random graphs with prescribed degrees, Internet Mathematics, 2011, 6(4): 489–522.

    Article  MathSciNet  MATH  Google Scholar 

  9. Goodman D, Rural Europe redux? Reflections on alternative agro-food networks and paradigm change, Sociologia Ruralis, 2004, 44(1): 3–16.

    Article  Google Scholar 

  10. Seyfang G, Ecological citizenship and sustainable consumption: Examining local organic food networks, Journal of Rural Studies, 2006, 22(4): 383–395.

    Article  Google Scholar 

  11. Goldenberg A, Zheng A X, and Fienberg S E, A survey of statistical network models, Foundations and Trends in Machine Learning, 2010, 2(2): 129–233.

    Article  MATH  Google Scholar 

  12. Fienberg S E, A brief history of statistical models for network analysis and open challenges, Journal of Computational and Graphical Statistics, 2012, 21(4): 825–839.

    Article  MathSciNet  Google Scholar 

  13. Krivitsky P N, Exponential-family random graph models for valued networks, Electronic Journal of Statistics, 2012, 6: 1100–1128.

    Article  MathSciNet  MATH  Google Scholar 

  14. Holland P W and Leinhardt S, An exponential family of probability distributions for directed graphs, Journal of the American Statistical Association, 1981, 76(373): 33–50.

    Article  MathSciNet  MATH  Google Scholar 

  15. Onnela J P, Saramäki J, Hyvönen J, et al., Analysis of a large-scale weighted network of one-toone human communication, New Journal of Physics, 2007, 9(6): 179.

    Article  Google Scholar 

  16. Calin O and Udrişte C, Maximum Entropy Distributions, Geometric Modeling in Probability and Statistics, Springer International Publishing, 2014, 165–187.

    Google Scholar 

  17. Sadinle M, The strength of arcs and edges in interaction networks: Elements of a model-based approach, arXiv preprint arXiv:1211.0312, 2012.

    Google Scholar 

  18. Chatterjee S, Diaconis P, and Sly A, Random graphs with a given degree sequence, The Annals of Applied Probability, 2011, 21(4): 1400–1435.

    Article  MathSciNet  MATH  Google Scholar 

  19. Yan T and Xu J, A central limit theorem in the ß-model for undirected random graphs with a diverging number of vertices, Biometrika, 2013, 100: 519–524.

    Article  MathSciNet  MATH  Google Scholar 

  20. Hillar C and Wibisono A, Maximum entropy distributions on graphs, arXiv preprint arXiv: 1301.3321, 2013.

    Google Scholar 

  21. Yan T, Zhao Y, and Qin H, Asymptotic normality in the maximum entropy models on graphs with an increasing number of parameters, Journal of Multivariate Analysis, 2015, 133: 61–76.

    Article  MathSciNet  MATH  Google Scholar 

  22. Yan T, Qin H, and Wang H, Asymptotics in undirected random graph models parameterized by the strengths of vertices, Statistica Sinica, 2016, 26: 273–293.

    MathSciNet  MATH  Google Scholar 

  23. Winship C and Mare R D, Regression models with ordinal variables, American Sociological Review, 1984, 48(4):: 512–525.

    Article  Google Scholar 

  24. Ananth C V and Kleinbaum D G, Regression models for ordinal responses: A review of methods and applications, International Journal of Epidemiology, 1997, 26(6): 1323–1333.

    Article  Google Scholar 

  25. Williams R, Ordered logit models-overview, 2015. Available at https://www3.nd.edu/ rwilliam/ stats3/L11.pdf.

    Google Scholar 

  26. Frank O and Strauss D, Markov graphs, Journal of the American Statistical Association, 1986, 81(395): 832–842.

    Article  MathSciNet  MATH  Google Scholar 

  27. Schweinberger M, Instability, sensitivity, and degeneracy of discrete exponential families, Journal of the American Statistical Association, 2011, 106(496): 1361–1370.

    Article  MathSciNet  MATH  Google Scholar 

  28. Snijders T A B, Pattison P E, and Robins G L, New specifications for exponential random graph models, Sociological Methodology, 2006, 36(1): 99–153.

    Article  Google Scholar 

  29. Pothoven K, Real and Functional Analysis, Springer Science & Business Media, 2013.

    MATH  Google Scholar 

  30. Hoeffding W, Probability inequalities for sums of bounded random variables, Journal of the American Statistical Association, 1963, 58(301): 13–30.

    Article  MathSciNet  MATH  Google Scholar 

  31. Loève M, Probability Theory II, Graduate Texts in Mathematics, 1994.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Statistics and Hubei Key Laboratory of Mathematical Sciences, Central China Normal University, Wuhan, 430079, China

    Wenlong Li, Ting Yan, Mohamed Abd Elgawad & Hong Qin

  2. Department of Mathematics, Faculty of Science, Benha University, Benha, 13518, Egypt

    Mohamed Abd Elgawad

Authors
  1. Wenlong Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ting Yan
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Mohamed Abd Elgawad
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Hong Qin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ting Yan.

Additional information

Qin’s research is partially supported by the National Natural Science Foundation of China under Grant Nos. 11271147, 11471135, and Yan’s research is partially supported by the National Natural Science Foundation of China under Grant No. 11401239, and the Self-Determined Research Funds of CCNU from the Colleges’s Basic Research and Operation of MOE (CCNU15A02032, CCNU15ZD011) and a Fund from KLAS (130026507).

This paper was recommended for publication by Editor SHAO Jun.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Li, W., Yan, T., Abd Elgawad, M. et al. Degree-based moment estimation for ordered networks. J Syst Sci Complex 30, 721–733 (2017). https://doi.org/10.1007/s11424-017-5307-5

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11424-017-5307-5

Keywords

Search

Navigation