  Motivated by growing concerns over ensuring privacy on social networks, we
develop new algorithms and impossibility results for fitting complex
statistical models to network data subject to rigorous privacy guarantees. We
consider the so-called node-differentially private algorithms, which compute
information about a graph or network while provably revealing almost no
information about the presence or absence of a particular node in the graph.
  We provide new algorithms for node-differentially private estimation for a
popular and expressive family of network models: stochastic block models and
their generalization, graphons. Our algorithms improve on prior work, reducing
their error quadratically and matching, in many regimes, the optimal nonprivate
algorithm. We also show that for the simplest random graph models ($G(n,p)$ and
$G(n,m)$), node-private algorithms can be qualitatively more accurate than for
more complex models---converging at a rate of $\frac{1}{\epsilon^2 n^{3}}$
instead of $\frac{1}{\epsilon^2 n^2}$. This result uses a new extension lemma
for differentially private algorithms that we hope will be broadly useful.
