In this paper, we focus on differential privacy preserving spectral graph analysis. Spectral graph analysis deals with the analysis of the spectra (eigenvalues and eigenvector components) of the graph's adjacency matrix or its variants. We develop two approaches to computing the e-differential eigen decomposition of the graph's adjacency matrix. The first approach, denoted as LNPP, is based on the Laplace Mechanism that calibrates Laplace noise on the eigenvalues and every entry of the eigenvectors based on their sensitivities. We derive the global sensitivities of both eigenvalues and eigenvectors based on the matrix perturbation theory. Because the output eigenvectors after perturbation are no longer orthogonormal, we postprocess the output eigenvectors by using the state-of-the-art vector orthogonalization technique. The second approach, denoted as SBMF, is based on the exponential mechanism and the properties of the matrix Bingham-von Mises-Fisher density for network data spectral analysis. We prove that the sampling procedure achieves differential privacy. We conduct empirical evaluation on a real social network data and compare the two approaches in terms of utility preservation (the accuracy of spectra and the accuracy of low rank approximation) under the same differential privacy threshold. Our empirical evaluation results show that LNPP generally incurs smaller utility loss.
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