  Online social networks are being increasingly used for analyzing various
societal phenomena such as epidemiology, information dissemination, marketing
and sentiment flow. Popular analysis techniques such as clustering and
influential node analysis, require the computation of eigenvectors of the real
graph's adjacency matrix. Recent de-anonymization attacks on Netflix and AOL
datasets show that an open access to such graphs pose privacy threats. Among
the various privacy preserving models, Differential privacy provides the
strongest privacy guarantees.
  In this paper we propose a privacy preserving mechanism for publishing social
network graph data, which satisfies differential privacy guarantees by
utilizing a combination of theory of random matrix and that of differential
privacy. The key idea is to project each row of an adjacency matrix to a low
dimensional space using the random projection approach and then perturb the
projected matrix with random noise. We show that as compared to existing
approaches for differential private approximation of eigenvectors, our approach
is computationally efficient, preserves the utility and satisfies differential
privacy. We evaluate our approach on social network graphs of Facebook, Live
Journal and Pokec. The results show that even for high values of noise variance
sigma=1 the clustering quality given by normalized mutual information gain is
as low as 0.74. For influential node discovery, the propose approach is able to
correctly recover 80 of the most influential nodes. We also compare our results
with an approach presented in [43], which directly perturbs the eigenvector of
the original data by a Laplacian noise. The results show that this approach
requires a large random perturbation in order to preserve the differential
privacy, which leads to a poor estimation of eigenvectors for large social
networks.

Cryptography and Security;Social and Information Networks;Physics and Society