April 2024 Edge differentially private estimation in the β-model via jittering and method of moments
Jinyuan Chang, Qiao Hu, Eric D. Kolaczyk, Qiwei Yao, Fengting Yi
Author Affiliations +
Ann. Statist. 52(2): 708-728 (April 2024). DOI: 10.1214/24-AOS2365


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.

Funding Statement

J. Chang and Q. Hu were supported in part by the National Natural Science Foundation of China (Grant nos. 71991472, 72125008 and 12326360).
J. Chang was also supported by the Center of Statistical Research at Southwestern University of Finance and Economics.
E. D. Kolaczyk was supported in part by U.S. National Science Foundation award SES-2120115 and Canadian NSERC RGPIN-2023-03566 and NSERC DGDND-2023-03566 awards.
Q. Yao was supported in part by the U.K. Engineering and Physical Sciences Research Council (Grant nos. EP/V007556/1 and EP/X002195/1).
F. Yi was supported in part by the National Key R&D Program of China (Grant no. 2022YFA1003701), the National Natural Science Foundation of China (Grant nos. 12071416, 12271472 and 12301363), the Fundamental Research Funds for the Central Universities (Grant no. JBK2101013), and the China Postdoctoral Science Foundation (Grant no. 2021M692663).


The authors are grateful to the Editor, an Associate Editor and three referees for their helpful suggestions. The authors also thank Vishesh Karwa for sharing code for differentially maximum likelihood estimation in the β-model.


Download Citation

Jinyuan Chang. Qiao Hu. Eric D. Kolaczyk. Qiwei Yao. Fengting Yi. "Edge differentially private estimation in the β-model via jittering and method of moments." Ann. Statist. 52 (2) 708 - 728, April 2024. https://doi.org/10.1214/24-AOS2365


Received: 1 December 2021; Revised: 1 January 2024; Published: April 2024
First available in Project Euclid: 9 May 2024

Digital Object Identifier: 10.1214/24-AOS2365

Primary: 62F12 , 68P27
Secondary: 91D30

Keywords: adaptive inference , bootstrap inference , data privacy , data release mechanism , edge differential privacy , phase transition , β-model

Rights: Copyright © 2024 Institute of Mathematical Statistics


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Vol.52 • No. 2 • April 2024
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