Limit results for estimation of connectivity matrix in multi-layer stochastic block models

Multi-layer networks arise naturally in various domains including biology, finance and sociology, among others. The multi-layer stochastic block model (multi-layer SBM) is commonly used for community detection in the multi-layer networks. Most of current literature focuses on statistical consistency of community detection methods under multi-layer SBMs. However, the asymptotic distributional properties are also indispensable which play an important role in statistical inference. In this work, we aim to study the estimation and asymptotic properties of the layer-wise scaled connectivity matrices in the multi-layer SBM. We study and analyze a computationally tractable method for estimating the scaled connectivity matrices. Under the multi-layer SBM and its variant multi-layer degree-corrected SBM, we establish the asymptotic normality of the estimated matrices under mild conditions, which can be used for interval estimation and hypothesis testing. Simulations show the superior performance of proposed method over existing methods in two considered statistical inference tasks. We apply the method to a real dataset and obtain interpretable results. In addition, we develop a moment estimator for the non-scaled connectivity matrices and study its asymptotic properties.