Edge differentially private estimation in the $\beta$-model via jittering and method of moments

TL;DR

This paper investigates the trade-off between privacy and statistical efficiency in estimating network parameters under the $eta$-model with edge differential privacy, introducing a method-of-moments estimator that adapts across privacy regimes.

Contribution

It proposes a novel estimator for the $eta$-model under differential privacy, revealing phase transitions in its behavior and developing an adaptive bootstrap for uniform inference across regimes.

Findings

Estimator exhibits phase transition depending on privacy level

Bootstrap method enables simultaneous inference of all parameters

Proposed method outperforms MLE in speed and memory

Abstract

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 $\beta$-model (Chatterjee, Diaconis and Sly, 2011) for edge differentially private network data released via jittering (Karwa, Krivitsky and Slavkovi\'{c}, 2017). Unlike most previous approaches based on maximum likelihood estimation for this network model, we proceed via 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 $\beta$-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.