HM-Modularity: A Harmonic Motif Modularity Approach for Multi-Layer Network Community Detection

Multi-layer network community detection has drawn an increasing amount of attention recently. Despite success, the existing methods mainly focus on the lower-order connectivity structure at the level of individual nodes and edges. And the higher-order connectivity structure has been largely ignored, which contains better signature of community compared with edges. The main challenges in utilizing higher-order structure for multi-layer network community detection are that the most representative higher-order structure may vary from one layer to another and the connectivity structure formed by the same node subset may exhibit different higher-order connectivity patterns in different layers. To this end, this paper proposes a novel higher-order structure, termed harmonic motif, which is a dense subgraph having on average the largest statistical significance in each layer. Based on the harmonic motif, a primary layer is constructed by integrating higher-order structural information from all layers. Additionally, the higher-order structural information of each individual layer is taken as the auxiliary information. A coupling is established between the primary layer and each auxiliary layer. Accordingly, a harmonic motif modularity is designed to generate the community structure. Extensive experiments on eleven real-world multi-layer network datasets have been conducted to confirm the effectiveness of the proposed method.