Motif adjacency matrix and spectral clustering of directed weighted networks

In the spectral clustering methods, different from the network division based on edges, some research has begun to divide the network based on network motifs; the corresponding objective function of partition also becomes related to the motif information. But, the related research on the directed weighted network needs to be further deepened. The weight of the network has a great influence on the structural attributes of the network, so it is necessary to extend the motif-based clustering to the weighted network. In this paper, a motif-based spectral clustering method for directed weighted networks is proposed. At the same time, this paper supplements the method of obtaining matrix expressions of the motif adjacency matrix in directed unweighted networks and provides a method to deal with the weight of networks, which will be helpful for the application research of motifs. This clustering method takes into account the higher-order connectivity patterns in networks and broadens the applicable range of spectral clustering to directed weighted networks. In this method, the motif-based clustering of directed weighted networks can be transformed into the clustering of the undirected weighted network corresponding to the motif-based adjacency matrix. The results show that the clustering method can correctly identify the partition structure of the benchmark network, and experiments on some real networks show that this method performs better than the method that does not consider the weight of networks.