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 SBMs. We develop a novel and efficient method to estimate 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 also apply the method to a real dataset and obtain interpretable results.

翻译：多层网络在生物学、金融学和社会学等多个领域自然涌现。多层随机块模型（multi-layer SBM）常用于多层网络中的社区检测。现有文献大多关注多层SBM下社区检测方法的统计一致性，然而渐近分布性质同样不可或缺，在统计推断中扮演重要角色。本研究旨在探讨多层SBM中层间缩放连通矩阵的估计及其渐近性质。我们提出了一种新颖高效的方法来估计缩放连通矩阵。在多层SBM及其变体多层度校正SBM框架下，我们在温和条件下建立了估计矩阵的渐近正态性，该性质可用于区间估计与假设检验。仿真实验表明，在两项统计推断任务中，所提方法性能优于现有方法。我们还将该方法应用于真实数据集，获得了可解释的结果。