Previous Articles    

Asymptotic in the Ordered Networks with a Noisy Degree Sequence

LUO Jing1, QIN Hong2   

  1. 1. Department of Mathematics and Statistics, South-Central University for Nationalities, Wuhan 430074, China;2. Department of Statistics, Zhongnan University of Economics and Law, Wuhan 430073, China
  • Received:2020-10-09 Revised:2020-12-14 Published:2022-06-20
  • Supported by:
    This research was supported by the Major Program of National Natural Science Foundation of China under Grant Nos. 61690210 and 61690212, the National Natural Science Foundation of China under Grant No. 61333003, and also by the Science Center Program of the National Natural Science Foundation of China under Grant No. 62188101.

LUO Jing, QIN Hong. Asymptotic in the Ordered Networks with a Noisy Degree Sequence[J]. Journal of Systems Science and Complexity, 2022, 35(3): 1137-1153.

In the case of the differential privacy under the Laplace mechanism, the asymptotic properties of parameter estimators have been derived in some special network models with common binary values, but the asymptotic properties in network models with the ordered values are lacking. In this paper, the authors release the degree sequences of the ordered networks under a general noisy mechanism with the discrete Laplace mechanism as a special case. The authors establish the asymptotic result including the consistency and asymptotical normality of the parameter estimator when the number of parameters goes to infinity. Simulations and a real data example are provided to illustrate asymptotic results.
[1] Zhou B, Pei J, and Luk W S, A brief survey on anonymization techniques for privacy preserving publishing of social network data, Acm Sigkdd Explorations Newsletter, 2008, 10(2):12-22.
[2] Yuan M, Lei C, and Yu P S, Personalized privacy protection in social networks, Proceedings of the Vldb Endowment, 2011, 4(2):141-150.
[3] Cutillo L A, Molva R, and Strufe T, Privacy preserving social networking through decentralization, International Conference on Wireless on-Demand Network Systems and Services, 2010.
[4] Lu W and Miklau G, Exponential random graph estimation under differential privacy, Proceedings of the 20th ACM SIGKDD International Conference on Knowlege Discovery and Data Mining, 2014.
[5] 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.
[6] Albert R and Barabási A, Statistical mechanics of complex networks, Review of Modern Physics, 2002, 74(1):47-97.
[7] Britton T, Deijfen M, and Martin-Löf A, Generating simple random graphs with prescribed degree distribution, Journal of Statistical Physics, 2006, 124(6):1377-1397.
[8] 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.
[9] Zhao Y, Levina E, and Zhu J, Consistency of community detection in networks under degreecorrected stochastic block models, The Annals of Statistics, 2012, 40(4):2266-2292.
[10] Hillar C and Wibisono A, Maximum entropy distributions on graphs, Avaible at:http://arxiv. org/abs/1301.3321, 2013.
[11] Chatterjee S, Diaconis P, and Sly A, Random graphs with a given degree sequence, The Annals of Applied Probability, 2011, 21(4):1400-1435.
[12] Blitzstein J and Diaconis P, A sequential importance sampling algorithm for generating random graphs with prescribed degrees, Internet Mathematics, 2011, 6(4):489-522.
[13] Rinaldo A, Petrović S, Fienberg S E, et al., Maximum lilkelihood estimation in the β-model, The Annals of Statistics, 2013, 41(3):1085-1110.
[14] Yan T and 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.
[15] Graham B S, An econometric model of network formation with degree heterogeneity, Econometrica, 2017, 85(4):1033-1063.
[16] Dzemski A, An empirical model of dyadic link formation in a network with unobserved heterogeneity, Working Papers in Economics, 2017.
[17] 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.
[18] Su L, Qian X, and Yan T, A note on a network model with degree heterogeneity and homophily, Statistics and Probability Letters, 2018, 138:27-30.
[19] Yan T, Jiang B, Fienberg S E, et al., Statistical inference in a directed network model with covariates, Journal of the American Statistical Association, 2019, 114(526):857-868.
[20] Li W, Yan T, Mohamed A E, et al., Degree-based moment estimation for ordered networks,Journal of Systems Science and Complexity, 2017, 30(3):721-733.
[21] Dwork C M F N K and Smith A, Calibrating noise to sensitivity in private data analysis, Proceedings od the 3rd Theory of Cryptography Conference, 2006, 265-284.
[22] Karwa V and Slavković A, Inference using noisy degrees:Differentially private beta-model and synthetic graphs, The Annals of Statistics, 2016, 44(1):87-112.
[23] Pan L and Yan T, Asymptotics in the β-model for networks with a differentially private degree sequence, Communications in Statistics-Theory and Methods, 2019, 49(18):4378-4393.
[24] Inusah S and Kozubowski T J, A discrete analogue of the laplace distribution, Journal of Statal Planning and Inference, 2006, 136(3):1090-1102.
[25] Zhang H and Chen S X, Concentration inequalities for statistical inference, arXiv:2011.02258, 2020.
[26] 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.
[27] McCullagh P, Regression models for ordinal data, Journal of the Royal Statistical Society:Series B (Methodological), 1980, 42(2):109-127.
[28] Bürkner P C and Vuorre M, Ordinal regression models in psychology:A tutorial, Advances in Methods and Practices in Psychological Science, 2019, 2(1):77-101.
[29] Hay M, Chao L, Miklou G, et al., Accurate estimation of the degree distribution of private networks, 9th IEEE International Conference on Data Mining, 2009, 169-178.
[30] Vershynin R, Introduction to the Non-Asymptotic Analysis of Random Matrices, Compressed Sensing, Theory and Applications, Cambridge University Press, Cambridge, 2012.
[31] Gautschi W, Some elementary inequalities relating to the gamma and incomplete gamma function, Journal of Mathematics and Physics, 1959, 38(1):77-81.
[32] 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.
[33] Gragg W and Tapia R, Optimal error bounds for the newton-kantorovich theorem, SIAM Journal on Numerical Analysis, 1974, 11(1):10-13.
[34] Chung K L and Zhong K, A Course in Probability Theory, Academic Press, New York, 2001.
[35] Lang S, Real and Functional Analysis, Springer-Verlag, New York, 1993.
[36] Hoeffding W, Probability inequalities for sums of bounded random variables, Journal of the American Statistical Association, 1963, 58(301):13-30.
[1] PENG Yunjie, ZHENG Xiaoqian, YU Wei, HE Kaixin, WANG Xuejun. Strong Law of Large Numbers for Weighted Sums of Random Variables and Its Applications in EV Regression Models [J]. Journal of Systems Science and Complexity, 2022, 35(1): 342-360.
[2] WAN Daqing. Exponential Sums over Finite Fields [J]. Journal of Systems Science and Complexity, 2021, 34(4): 1225-1278.
[3] JIANG Yanguang,HUANG Yi,XUE Wenchao,FANG Haitao. On Designing Consistent Extended Kalman Filter [J]. Journal of Systems Science and Complexity, 2017, 30(4): 751-764.
[4] ZHAO Jiwei,SHAO Jun. Approximate Conditional Likelihood for Generalized Linear Models with General Missing Data Mechanism [J]. Journal of Systems Science and Complexity, 2017, 30(1): 139-153.
[5] FAN Yan,GAI Yujie,ZHU Lixing. Asymtotics of Dantzig Selector for a General Single-Index Model [J]. Journal of Systems Science and Complexity, 2016, 29(4): 1123-1144.
[6] LI Danping,RONG Ximin,ZHAO Hui. Time-Consistent Investment Strategy for DC Pension Plan with Stochastic Salary Under CEV Model [J]. Journal of Systems Science and Complexity, 2016, 29(2): 428-454.
[7] JIN Qingfang,LIU Zhuojun, WU Baofeng, ZHANG Xiaoming. A Combinatorial Condition and Boolean Functions with Optimal Algebraic Immunity [J]. Journal of Systems Science and Complexity, 2015, 28(3): 725-742.
[8] HUANG Rui, CUI Hengjian. Consistency of Chi-Squared Test with Varying Number of Classes [J]. Journal of Systems Science and Complexity, 2015, 28(2): 439-450.
[9] JIA Shihui , WAN Zhongping. A PENALTY FUNCTION METHOD FOR SOLVING ILL-POSED BILEVEL PROGRAMMING PROBLEM VIA WEIGHTED SUMMATION [J]. Journal of Systems Science and Complexity, 2013, 26(6): 1019-1027.
[10] YUAN Yuan , YOU Jinhong , ZHOU Yong. EFFICIENT ESTIMATION OF SEEMINGLY UNRELATED ADDITIVE NONPARAMETRIC REGRESSION MODELS [J]. Journal of Systems Science and Complexity, 2013, 26(4): 595-608.
[12] Sili NIU ,Yamei LIU. CLT OF WAVELET ESTIMATOR IN SEMIPARAMETRIC MODEL WITH CORRELATED ERRORS [J]. Journal of Systems Science and Complexity, 2012, 25(3): 567-581.
[13] Huixiu ZHAO, Jinguan LIN. THE LARGE SAMPLE PROPERTIES OF THE SOLUTIONS OF GENERAL ESTIMATING EQUATIONS [J]. Journal of Systems Science and Complexity, 2012, 25(2): 315-328.
[14] Qunying WU. FURTHER STUDY STRONG CONSISTENCY OF$M$ ESTIMATOR IN LINEAR  MODEL FOR $\widetilde{\rho}$-MIXING RANDOM SAMPLES [J]. Journal of Systems Science and Complexity, 2011, 24(5): 969-980.
[15] Hanying LIANG. ASYMPTOTIC NORMALITY OF WAVELET ESTIMATOR IN HETEROSCEDASTIC MODEL WITH $\alpha$-MIXING ERRORS [J]. Journal of Systems Science and Complexity, 2011, 24(4): 725-737.
Full text