Katz Fractal Dimension of Geoelectric Field during Severe Geomagnetic Storms

We are concerned with the time series resulting from the computed local horizontal geoelectric field, obtained with the aid of a 1-D layered Earth model based on local geomagnetic field measurements, for the full solar magnetic cycle of 1996–2019, covering the two consecutive solar activity cycles 23 and 24. To our best knowledge, for the first time, the roughness of severe geomagnetic storms is considered by using a monofractal time series analysis of the Earth electric field. We show that during severe geomagnetic storms the Katz fractal dimension of the geoelectric field grows rapidly.

Determination of multifractal dimensions of complex networks by means of the sandbox algorithm

Complex networks have attracted much attention in diverse areas of science and technology. Multifractal analysis (MFA) is a useful way to systematically describe the spatial heterogeneity of both theoretical and experimental fractal patterns. In this paper, we employ the sandbox (SB) algorithm proposed by Tél et al. (Physica A 159, 155-166 (1989)), for MFA of complex networks. First, we compare the SB algorithm with two existing algorithms of MFA for complex networks: the compact-box-burning algorithm proposed by Furuya and Yakubo (Phys. Rev. E 84, 036118 (2011)), and the improved box-counting algorithm proposed by Li et al. (J. Stat. Mech.: Theor. Exp. 2014, P02020 (2014)) by calculating the mass exponents τ(q) of some deterministic model networks. We make a detailed comparison between the numerical and theoretical results of these model networks. The comparison results show that the SB algorithm is the most effective and feasible algorithm to calculate the mass exponents τ(q) and to explore the multifractal behavior of complex networks. Then, we apply the SB algorithm to study the multifractal property of some classic model networks, such as scale-free networks, small-world networks, and random networks. Our results show that multifractality exists in scale-free networks, that of small-world networks is not obvious, and it almost does not exist in random networks.