  Modern multi-layer networks are commonly stored and analyzed in a local and
distributed fashion because of the privacy, ownership, and communication costs.
The literature on the model-based statistical methods for community detection
based on these data is still limited. This paper proposes a new method for
consensus community detection and estimation in a multi-layer stochastic block
model using locally stored and computed network data with privacy protection. A
novel algorithm named privacy-preserving Distributed Spectral Clustering
(ppDSC) is developed. To preserve the edges' privacy, we adopt the randomized
response (RR) mechanism to perturb the network edges, which satisfies the
strong notion of differential privacy. The ppDSC algorithm is performed on the
squared RR-perturbed adjacency matrices to prevent possible cancellation of
communities among different layers. To remove the bias incurred by RR and the
squared network matrices, we develop a two-step bias-adjustment procedure. Then
we perform eigen-decomposition on the debiased matrices, aggregation of the
local eigenvectors using an orthogonal Procrustes transformation, and k-means
clustering. We provide theoretical analysis on the statistical errors of ppDSC
in terms of eigen-vector estimation. In addition, the blessings and curses of
network heterogeneity are well-explained by our bounds.
