A standing challenge in data privacy is the trade-off between the level of privacy and the efficiency of statistical inference. Here, we conduct an in-depth study of this trade-off for parameter estimation in the β-model (Ann. Appl. Probab. 21 (2011) 1400–1435) for edge differentially private network data released via jittering (J. R. Stat. Soc. Ser. C. Appl. Stat. 66 (2017) 481–500). Unlike most previous approaches based on maximum likelihood estimation for this network model, we proceed via the method of moments. This choice facilitates our exploration of a substantially broader range of privacy levels—corresponding to stricter privacy—than has been to date. Over this new range, we discover our proposed estimator for the parameters exhibits an interesting phase transition, with both its convergence rate and asymptotic variance following one of three different regimes of behavior depending on the level of privacy. Because identification of the operable regime is difficult, if not impossible in practice, we devise a novel adaptive bootstrap procedure to construct uniform inference across different phases. In fact, leveraging this bootstrap we are able to provide for simultaneous inference of all parameters in the β-model (i.e., equal to the number of nodes), which, to our best knowledge, is the first result of its kind. Numerical experiments confirm the competitive and reliable finite sample performance of the proposed inference methods, next to a comparable maximum likelihood method, as well as significant advantages in terms of computational speed and memory.

Publication:

Annals of Statistics. 52(2): 708-728 (April 2024)

http://dx.doi.org/10.1214/24-AOS2365

Author:

Jinyuan Chang

Joint Laboratory of Data Science and Business Intelligence, Southwestern University of Finance and Economics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences

Email: changjinyuan@amss.ac.cn

Qiao Hu

Joint Laboratory of Data Science and Business Intelligence, Southwestern University of Finance and Economics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences

Eric D. Kolaczyk

Department of Mathematics and Statistics, McGill University

Qiwei Yao

Department of Statistics, London School of Economics and Political Science

Fengting Yi

Yunnan Key Laboratory of Statistical Modeling and Data Analysis, Yunnan University