Accuracy of the ball-covering approach for fractal dimensions of complex networks and a rank-driven algorithm.

Minimum box-covering method is a basic tool to measure fractal dimension of a network but unfortunately belongs to a family of NP-hard problems. Finding more accurate approximation in a relative shorter time is an important problem and attracts considerable attention. In this paper, the accuracy of the ball-covering approach for fractal dimension is measured by the relative error eta to the true fractal dimension, and its upper bound epsilon is analytically deduced by the linear least-squares method. When the ball-covering algorithm is applied to real-world networks, the relative error eta always stays in a narrow range which is far below its theoretical upper bound epsilon . It indicates that the ball-covering approach provides a helpful method to approximate the true fractal dimension. Then a rank-driven ball-covering algorithm is presented. It is demonstrated that the ranking mechanism ensures the search steps of our algorithm for finding balls are obviously less than that of the original algorithm introduced by Kim [Chaos 17, 026116 (2007)] and our algorithm is much more efficient.