Community Detection in Fully-Connected Multi-layer Networks Through Joint Nonnegative Matrix Factorization

Modern data analysis and processing tasks typically involve large sets of structured data. Graphs provide a powerful tool to describe the structure of such data, where the entities and the relationships between them are modeled as the nodes and edges of the graph. Traditional single layer network models are insufficient for describing the multiple entity types and modes of interaction encountered in real-world applications. Recently, multi-layer network models, which consider the different types of interactions both within and across layers, have emerged to model these systems. One of the important tools in understanding the topology of these high-dimensional networks is community detection. In this paper, a joint nonnegative matrix factorization approach is proposed to detect the community structure in multi-layer networks. The proposed approach models the multi-layer network as the union of a multiplex and bipartite network and formulates community detection as a regularized optimization problem. This optimization problem simultaneously finds the nonnegative low-rank embedding of the intra- and inter-layer adjacency matrices while minimizing the distance between the two to guarantee pair-wise similarity across embeddings. The proposed approach can detect the community structure for both homogeneous and heterogeneous multi-layer networks and is robust to noise and sparsity. The performance of the proposed approach is evaluated for both simulated and real networks and compared to state-of-the-art methods.