Motif-based mix-order nonnegative matrix factorization for community detection

Community structure is one of the important characteristics of complex networks, so it is of great application value to correctly detect community structure in the study of network structure. Nonnegative matrix factorization (NMF) has been proved to be an ideal model of the community detection. Traditional NMF only focuses on the first-order structure (such as adjacency matrix), but does not consider the higher-order structure (such as motif adjacency matrix). However, only considering one of them cannot well represent the global structural characteristics of complex networks. In this paper, we propose a new Mixed-Order Nonnegative Matrix Factorization (MONMF) framework, which can model both first-order and higher-order structures. Previous nonnegative matrix factorization is mostly used in undirected networks, but we will study based on a variety of motif types in directed networks, use motifs to capture higher-order structures in networks, and introduce linear and nonlinear methods to combine the adjacency matrix representing the first-order structure with the motif adjacency matrix representing the higher-order structure to construct a new feature matrix of NMF. At the same time, we introduce the missing edge matrix that characterizes the edgeless connection structure of the network, and gives the expression of the motif adjacency matrix of the three-node open simple motif and the three-node open anchor motif. The MONMF operation is mainly performed on different real networks for open simple motifs and open anchor motifs. Compared with the comparison methods, MONMF can significantly improve the performance of community detection in complex networks.