  The goal of privacy-preserving social graph publishing is to protect
individual privacy while preserving data utility. Community structure, which is
an important global pattern of nodes, is a crucial data utility as it serves as
fundamental operations for many graph analysis tasks. Yet, most existing
methods with differential privacy (DP) commonly fall in edge-DP to sacrifice
security in exchange for utility. Moreover, they reconstruct graphs from the
local feature-extraction of nodes, resulting in poor community preservation.
Motivated by this, we propose PrivCom, a strict node-DP graph publishing
algorithm to maximize the utility on the community structure while maintaining
a higher level of privacy. Specifically, to reduce the huge sensitivity, we
devise a Katz index-based private graph feature extraction method, which can
capture global graph structure features while greatly reducing the global
sensitivity via a sensitivity regulation strategy. Yet, with a fixed
sensitivity, the feature captured by Katz index, which is presented in matrix
form, requires privacy budget splits. As a result, plenty of noise is injected,
thereby mitigating global structural utility. To this end, we design a private
Oja algorithm approximating eigen-decomposition, which yields the noisy Katz
matrix via privately estimating eigenvectors and eigenvalues from extracted
low-dimensional vectors. Experimental results confirm our theoretical findings
and the efficacy of PrivCom.
