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A standing challenge in data privacy is the trade-off between the level ofprivacy and the efficiency of statistical inference. Here we conduct anin-depth study of this trade-off for parameter estimation in the $\beta$-model(Chatterjee, Diaconis and Sly, 2011) for edge differentially private networkdata released via jittering (Karwa, Krivitsky and Slavkovi\'c, 2017). Unlikemost previous approaches based on maximum likelihood estimation for thisnetwork model, we proceed via method of moments. This choice facilitates ourexploration of a substantially broader range of privacy levels -- correspondingto stricter privacy -- than has been to date. Over this new range we discoverour proposed estimator for the parameters exhibits an interesting phasetransition, with both its convergence rate and asymptotic variance followingone of three different regimes of behavior depending on the level of privacy.Because identification of the operable regime is difficult to impossible inpractice, we devise a novel adaptive bootstrap procedure to construct uniforminference across different phases. In fact, leveraging this bootstrap we areable to provide for simultaneous inference of all parameters in the$\beta$-model (i.e., equal to the number of vertices), which would appear to bethe first result of its kind. Numerical experiments confirm the competitive andreliable finite sample performance of the proposed inference methods, next to acomparable maximum likelihood method, as well as significant advantages interms of computational speed and memory.

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