A Harmonic Motif Modularity Approach for Multi-layer Network Community Detection

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During the past several years, multi-layer network community detection has drawn an increasing amount of attention and many approaches have been developed from different perspectives. Despite the success, they mainly rely on the lower-order connectivity structure at the level of individual nodes and edges. However, the higher-order connectivity structure plays the essential role as the building block for multiplex networks, which may contain better signature of community than edge. The main challenge in utilizing higher-order structure for multi-layer network community detection is that the most representative higher-order structure may vary from one layer to another. In this paper, we propose a higher-order structural approach for multi-layer network community detection, termed harmonic motif modularity (HM-Modularity). The key idea is to design a novel higher-order structure, termed harmonic motif, which is able to integrate higher-order structural information from multiple layers to construct a primary layer. The higher-order structural information of each individual layer is also extracted, which is taken as the auxiliary information for discovering the multi-layer community structure. A coupling is established between the primary layer and each auxiliary layer. Finally, a harmonic motif modularity is designed to generate the community structure. By solving the optimization problem of the harmonic motif modularity, the community labels of the primary layer can be obtained to reveal the community structure of the original multi-layer network. Experiments have been conducted to show the effectiveness of the proposed method.