Multi-layer networks provide complementary views of interactions among the same set of nodes. In practice, layers with similar community structures may have substantially different network sparsity, causing sparsity variation to obscure the structural patterns of interest. To address this problem, a sparsity heterogeneous mixture multi-layer stochastic block model is proposed, which separates layer-specific sparsity from group-specific community structure. Based on the induced low-rank tensor representation, a sparsity-corrected tensor clustering method is proposed to remove the layer-specific scale effect before clustering, without explicitly estimating the sparsity parameters. Theoretical guarantees for tensor decomposition, layer clustering, and community detection are established. The results further characterize the range of sparsity heterogeneity allowed for consistent recovery, including regimes where the sparsity ratio diverges with the network dimensions. Numerical studies and an application to international trade networks from BACI demonstrate the effectiveness of the proposed method under heterogeneous layer sparsity.

翻译：暂无翻译