  The goal of community detection over graphs is to recover underlying
labels/attributes of users (e.g., political affiliation) given the connectivity
between users (represented by adjacency matrix of a graph). There has been
significant recent progress on understanding the fundamental limits of
community detection when the graph is generated from a stochastic block model
(SBM). Specifically, sharp information theoretic limits and efficient
algorithms have been obtained for SBMs as a function of $p$ and $q$, which
represent the intra-community and inter-community connection probabilities. In
this paper, we study the community detection problem while preserving the
privacy of the individual connections (edges) between the vertices. Focusing on
the notion of $(\epsilon, \delta)$-edge differential privacy (DP), we seek to
understand the fundamental tradeoffs between $(p, q)$, DP budget $(\epsilon,
\delta)$, and computational efficiency for exact recovery of the community
labels.
  To this end, we present and analyze the associated information-theoretic
tradeoffs for three broad classes of differentially private community recovery
mechanisms: a) stability based mechanism; b) sampling based mechanisms; and c)
graph perturbation mechanisms. Our main findings are that stability and
sampling based mechanisms lead to a superior tradeoff between $(p,q)$ and the
privacy budget $(\epsilon, \delta)$; however this comes at the expense of
higher computational complexity. On the other hand, albeit low complexity,
graph perturbation mechanisms require the privacy budget $\epsilon$ to scale as
$\Omega(\log(n))$ for exact recovery. To the best of our knowledge, this is the
first work to study the impact of privacy constraints on the fundamental limits
for community detection.
