Multi-layer networks are networks on a set of entities (nodes) with multiple types of relations (edges) among them where each type of relation/interaction is represented as a network layer. As with single layer networks, community detection is an important task in multi-layer networks. A large group of popular community detection methods in networks are based on optimizing a quality function known as the modularity score, which is a measure of presence of modules or communities in networks. Hence a first step in community detection is defining a suitable modularity score that is appropriate for the network in question. Here we introduce several multi-layer network modularity measures under different null models of the network, motivated by empirical observations in networks from a diverse field of applications. In particular we define the multi-layer configuration model, the multi-layer expected degree model and their various modifications as null models for multi-layer networks to derive different modularities. The proposed modularities are grouped into two categories. The first category, which is based on degree corrected multi-layer stochastic block model, has the multi-layer expected degree model as their null model. The second category, which is based on multi-layer extensions of Newman-Girvan modularity, has the multi-layer configuration model as their null model. These measures are then optimized to detect the optimal community assignment of nodes. We compare the effectiveness of the measures in community detection in simulated networks and then apply them to four real networks.
A Multilayer Probit Network Model for Community Detection with Dependent Edges and Layers
The Chinese University of Hong Kong（香港中文大学）；Southern University of Science and Technology（南方科技大学）；Shanghai Center for Mathematical Sciences, Fudan University（复旦大学上海数学中心）
Privacy-Preserving Community Detection for Locally Distributed Multiple Networks
School of Mathematics, Northwest University, China（中国西北大学数学学院）；University of Pennsylvania, U.S.A.（美国宾夕法尼亚大学）；School of Management, Xi’an Jiaotong University, China（中国西安交通大学管理学院）
