The asymptotic normality of the fixed number of the maximum likelihood estimators (MLEs) in the directed finite weighted network models with an increasing bi-degree sequence has been established recently. In this article, we further derive the central limit theorem for linear combinations of all the MLEs with an increasing dimension when the edges take finite discrete weight. Simulation studies are provided to illustrate the asymptotic results.
Jing LUO
,
Hong QIN
,
Zhenghong WANG
. ASYMPTOTIC DISTRIBUTION IN DIRECTED FINITE WEIGHTED RANDOM GRAPHS WITH AN INCREASING BI-DEGREE SEQUENCE[J]. Acta mathematica scientia, Series B, 2020
, 40(2)
: 355
-368
.
DOI: 10.1007/s10473-020-0204-8
[1] Agneessens F, Roose H, Waege H. Choices of theatre events:p* models for affiliation networks with attributes. Metodoloski Zvezki, 2004, 1:419-439
[2] Battiston S, Catanzaro M. Statistical properties of corporate board and director networks. The European Physical Journal B-Condensed Matter and Complex Systems, 2004, 38(2):345-352
[3] Bickel P J, Chen A, Levina E, et al. The method of moments and degree distributions for network models. The Annals of Statistics, 2011, 39(5):2280-2301
[4] Billingsley P. Convergence of probability measures. John Wiley & Sons, 1968
[5] Blitzstein J, Diaconis P. A sequential importance sampling algorithm for generating random graphs with prescribed degrees. Internet Mathematics, 2011, 6(4):489-522
[6] Britton T, Deijfen M, Martin-Löf A. Generating simple random graphs with prescribed degree distribution. Journal of Statistical Physics, 2006, 124(6):1377-1397
[7] Chatterjee S, Diaconis P, Sly A. Random graphs with a given degree sequence. The Annals of Applied Probability, 2011, 21(4):1400-1435
[8] Darmois G. Sur les lois de probabilitéa estimation exhaustive. CR Acad Sci Paris, 1935, 260(1265):85
[9] Ergün G. Human sexual contact network as a bipartite graph. Physica A:Statistical Mechanics and its Applications, 2002, 308(1):483-488
[10] 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
[11] Freeman L C, Webster C M, Kirke D M. Exploring social structure using dynamic three-dimensional color images. Social Networks, 1998, 20(2):109-118
[12] Freeman S, Freeman L. The networkers network:A study of the impact of a new communications medium on sociometric structure//Social Science Research Reports 46. Irvine, CA:University of California, 1979
[13] Hillar C, Wibisono A. Maximum entropy distributions on graphs. arXiv:1301.3321, 2013
[14] Holland P W, Leinhardt S. An exponential family of probability distributions for directed graphs. Journal of the American Statistical Association, 1981, 76(373):33-50
[15] Koopman B O. On distributions admitting a sufficient statistic. Transactions of the American Mathematical Society, 1936, 39(3):399-409
[16] Luo J, Qin H, Yan T, Zeyneb L. A note on asymptotic distributions in directed exponential random graph models with bi-degree sequences. Communications in Statistics-Theory and Methods, 2017, 46(18):8852-8864
[17] Newman M E. Scientific collaboration networks. I. network construction and fundamental results. Physical Review E, 2001, 64(1):016131
[18] Pitman E J G. Sufficient statistics and intrinsic accuracy//Mathematical Proceedings of the cambridge Philosophical society. Cambridge Univ Press, 1936, 32:567-579
[19] Rinaldo A, Petrović S, Fienberg S E. Maximum lilkelihood estimation in the beta-model. The Annals of Statistics, 2013, 41(3):1085-1110
[20] Skvoretz J, Faust K. Logit models for affiliation networks. Sociological Methodology, 2002, 29(1):253-280
[21] Snijders T A, Lomi A, Torló V J. A model for the multiplex dynamics of two-mode and one-mode networks, with an application to employment preference, friendship, and advice. Social Networks, 2013, 35(2):265-276
[22] Wang P, Sharpe K, Robins G L, Pattison P E. Exponential random graph (p*) models for affiliation networks. Social Networks, 2009, 31(1):12-25
[23] Watts D J, Strogatz S H. Collective dynamics of small-world networks. Nature, 1998, 393(6684):440-442
[24] Yan T, Leng C, Zhu J. Asymptotics in directed exponential random graph models with an increasing bi-degree sequence. The Annals of Statistics, 2016, 44(1):31-57
[25] Yan T, Qin H, Wang H. Asymptotics in undirected random graph models parameterized by the strengths of vertices. Statistica Sinica, 2016, 26(1):273-293
[26] Yan T, Xu J. A central limit theorem in the β-model for undirected random graphs with a diverging number of vertices. Biometrika, 2013, 100(2):519-524
[27] Yong Z, Chen S, Qin H, Yan T. Directed weighted random graphs with an increasing bi-degree sequence. Statistics and Probability Letters, 2016, 119:235-240
[28] Zhang H, Liu Y, Li B. Notes on discrete compound poisson model with applications to risk theory. Insurance Mathematics and Economics, 2014, 59:325-336
[29] Zhao Y, Levina E, Zhu J. Consistency of community detection in networks under degree-corrected stochastic block models. The Annals of Statistics, 2012, 40(4):2266-2292
