Due to extensive research on complex networks, fractal analysis with scale invariance is applied to measure the topological structure and self-similarity of complex networks. Fractal dimension can be used to quantify the fractal properties of the complex networks. However, in the existing box covering algorithms, accurately calculating the fractal dimension of complex networks is still an NP-hard problem. Therefore, in this paper, an improved overlapping box covering algorithm is proposed to explore a more accurate and effective method to calculate the fractal dimension of complex networks. Moreover, in order to verify the effectiveness of the algorithm, the improved algorithm is applied to six complex networks, and compared with other algorithms. Finally, the experimental results demonstrate that the improved overlapping box covering algorithm can cover the whole networks with fewer boxes. In addition, the improved overlapping box covering algorithm is a high accuracy and low time complexity method for calculating the fractal dimension of complex networks. INDEX TERMS Overlapping box covering algorithm, fractal dimension, complex networks.

In the research field of complex networks, the network nodes and edges are the fundamental indicators and often used as the preliminary step in the structural analysis of complex networks. In this paper, first, considering the differences in the node degree distribution, the traditional box-covering algorithm is modified by regarding the number of nodes in the box as the node degree, it is proved that multifractal characteristics are the nature of weighted complex networks. Moreover, multifractal spectrums of eight different networks are obtained and the variations in three eigenvalues corresponding are further analyzed, by comparing the existing and modified algorithms. Finally, we can make the practical conclusion that the multifractal characteristics of weighted complex networks are affected by the differences in the node degree distribution.INDEX TERMS Degree volume dimension, multifractal, modified box-covering algorithm, weighted complex networks.

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