Consistency of Privacy-Preserving Spectral Clustering under the Stochastic Block Model

The stochastic block model (SBM) -- a network model featuring community structure -- is often selected as the fundamental setting in which to analyze the theoretical properties of community detection methods. We consider the problem of privacy-preserving spectral clustering of SBMs under $\varepsilon$-edge differential privacy (DP) for networks, and offer practical interpretations from both the central-DP and local-DP perspectives. Using a randomized response privacy mechanism called the edge-flip mechanism, we take a first step toward theoretical analysis of differentially private community detection by demonstrating conditions under which this strong privacy guarantee can be upheld while achieving spectral clustering convergence rates that match the known rates without privacy. We prove the strongest theoretical results are achievable for dense networks (those with node degree linear in the number of nodes), while weak consistency is achievable under mild sparsity (node degree greater than $n^{-1/2}$). We empirically demonstrate our results on a number of network examples.