Relative entropy of k-order edge capacity for nodes similarity analysis

The solutions to many problems on complex networks depend on the calculation of the similarity between nodes. The existing methods face the problems of the lack of hierarchical information richness or large computational requirements. In order to flexibly analyze the similarity of nodes on an optional multi-order scale as needed, we propose a novel method for calculating the similarity based on the relative entropy of [Formula: see text]-order edge capacity in this paper. The distribution of edges affects the network heterogeneity, information propagation, node centrality and so on. Entropy of [Formula: see text]-order edge capacity can represent the edge distribution feature in the range of [Formula: see text]-order of node. It increases as [Formula: see text] increases and converges at the eccentricity of the node. Relative entropy of [Formula: see text]-order edge capacity can be used to compare the similarity of edge distribution between nodes within [Formula: see text]-order. As order [Formula: see text] increases, upper bound of the relative entropy possibly increases. Relative entropy gets the maximum when nodes compared with isolated nodes. By quantifying the effect difference of the most similar nodes on the network structure and information propagation, we compared relative entropy of [Formula: see text]-order edge capacity with some major similarity methods in the experiments, combined with visual analysis. The results show the rationality and effectiveness of the proposed method.