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.

翻译：我们研究边差分隐私条件下的谱图聚类问题。提出一种矩阵混洗机制，该机制将随机边翻转与邻接矩阵的随机排列相结合。虽然单独使用边翻转在图规模增长时只能提供恒定的$\varepsilon$保证，但混洗可以放大隐私保护效果，使得有效$\varepsilon$随节点数量增加而趋于零。我们构建了一个统一的误差分析框架——基于Davis-Kahan扰动理论与分类间隔界——该框架以隐私预算、特征间隙和社区数量的函数形式，给出了所考虑所有机制的显式误分类率。应用该框架后，我们证明矩阵混洗机制实现了$\tilde{O}(1/n)$量级的误分类率，这明显优于私有PCA文献中两种经典差分隐私基线方法：直接应用于邻接矩阵的高斯机制（Analyze Gauss）和噪声幂法，这两种方法在$n$上的量级均为$\tilde{O}(1)$。此外，我们还提出一种用于估计社区数量的私有特征间隙检测算法。在合成网络与真实网络上的实验验证了我们的理论发现。