  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.
