Multi‐layer networks are often used to represent multiple types of relationships between nodes in network studies. In this paper, we investigate the community detection problem in multi‐layer networks. Specifically, we consider the multi‐layer stochastic block model (MLSBM), which assumes that the community memberships are shared across all network layers, while other model parameters can be different between different layers. Variational methods have been developed to fit the MLSBM, but they do not scale well to very large networks. Inspired by the iterative pseudo‐likelihood maximization strategy for single networks, we develop a pseudo‐likelihood based algorithm to fit the MLSBM and estimate the community labels, and we also extend the proposed algorithm to the degree‐corrected case to deal with degree heterogeneity. The proposed algorithms are fast and can cope with multi‐layer networks with up to millions of nodes. The advantages of the proposed methods in both community detection and computational efficiency are demonstrated by numerical studies.

中文翻译：

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多层网络集成社区检测的快速算法

多层网络通常用于表示网络研究中节点之间的多种类型的关系。在本文中，我们研究了多层网络中的社区检测问题。具体来说，我们考虑多层随机块模型（MLSBM），该模型假设社区成员资格在所有网络层之间共享，而其他模型参数在不同层之间可能有所不同。已经开发出各种方法来适合MLSBM，但是它们不能很好地扩展到非常大的网络。受单网络迭代伪似然最大化策略的启发，我们开发了一种基于伪似然的算法来拟合MLSBM并估计社区标签，并且还将拟议算法扩展到经过度校正的情况下以处理度异质性。所提出的算法是快速的，并且可以应对具有多达数百万个节点的多层网络。数值研究证明了所提方法在社区检测和计算效率上的优势。