Modularity in complex multilayer networks with multiple aspects: a static perspective

Complex systems are usually illustrated by networks which capture the topology of the interactions between the entities. To better understand the roles played by the entities in the system, one needs to uncover the underlying community structure of the system. In recent years, systems with interactions that have various types or can change over time between the entities have attracted much research attention. However, algorithms aiming at solving the key problem—community detection—in multilayer networks are still limited. In this work, we first introduce the multilayer network model representation. Then based on this model, we naturally derive the multilayer modularity—a widely adopted objective function of community detection in networks—from a static perspective to evaluate the quality of the communities detected in multilayer networks. It enables us to better understand the essence of the modularity by pointing out the specific kind of communities that will lead to a high modularity score. We also propose a spectral method called mSpec for the optimization of the proposed modularity function based on the supra-adjacency representation of the multilayer networks. Experiments on the electroencephalograph network and the comparison results on several empirical multilayer networks demonstrate the feasibility and reliable performance of the proposed method.