Differentially Private Spectral Graph Clustering: Balancing Privacy, Accuracy, and Efficiency
We study spectral graph clustering under edge differential privacy. We propose a matrix shuffling mechanism that combines randomized edge flipping with a random permutation of the adjacency matrix. While edge flipping alone provides only a constant $\varepsilon$ guarantee as the graph grows, shuffling amplifies privacy so that the effective $\varepsilon$ tends to zero with the number of nodes. We develop a unified error analysis framework -- based on Davis--Kahan perturbation theory and a classification-margin bound -- that gives explicit misclassification rates for all the mechanisms considered as a function of the privacy budget, eigengap, and number of communities. Applying this framework, we show that the matrix shuffling mechanism achieves an error rate scaling of $\tilde{O}(1/n)$, a clear improvement over two canonical DP baselines from the private PCA literature: the Gaussian mechanism applied directly to the adjacency matrix (Analyze Gauss) and the noisy power method, both of which scale as $\tilde{O}(1)$ in $n$. We further propose a private spectral gap detection algorithm for estimating the number of communities. Experiments on synthetic and real-world networks validate our theoretical findings.