Moment-integrated Bias-adjusted Spectral Method for Community Detection in Multi-layer Networks

To detect the global community structure of multi-layer networks, individual layers might not provide sufficient information. It is of vital importance to effectively complement the information from the network data. In this paper, under the framework of multi-layer stochastic block model, a Spectral method with Moments integration and Bias Adjustment (SpecMBA) is proposed for community detection. The key distinguishing feature of SpecMBA is the adaptive integration of both the first and second moments of adjacency matrices with a hyperparameter α∈[−1,∞), which overcomes the limitations of fixed-form aggregation. Furthermore, SpecMBA adjusts for the bias caused by noise heteroskedasticity to mitigate signal distortion. In addition, a data-driven likelihood-based approach is proposed to select the optimal α. This adaptive aggregation ensures robust and competitive performance across diverse scenarios, which has been confirmed in the numerical studies. Under mild conditions, the community detection consistency for SpecMBA is established. The application on the international food trading network reveals interesting findings.