A new information dimension of complex networks

The fractal and self-similarity properties are revealed in many complex networks. The classical information dimension is an important method to study fractal and self-similarity properties of planar networks. However, it is not practical for real complex networks. In this Letter, a new information dimension of complex networks is proposed. The nodes number in each box is considered by using the box-covering algorithm of complex networks. The proposed method is applied to calculate the fractal dimensions of some real networks. Our results show that the proposed method is efficient when dealing with the fractal dimension problem of complex networks.     

Exterior dimension of fat fractals

Geometric scaling properties of fat fractal sets (fractals with finite volume) are discussed and characterized via the introduction of a new dimension-like quantity which we call the exterior dimension. In addition, it is shown that the exterior dimension is related to the “uncertainty exponent” previously used in studies of fractal basin boundaries, and it is shown how this connection can be exploited to determine the exterior dimension. Three illustrative applications are described, two in nonlinear dynamics and one dealing with blood flow in the body. Possible relevance to porous materials and ballistic driven aggregation is also noted.     

On spectral asymptotics for domains with fractal boundaries

We discuss the spectral properties of the Laplacian for domains Ω with fractal boundaries. The main goal of the article is to find the second term of spectral asymptotics of the counting functionN(λ) or its integral transformations: Θ-function, ξ-function. For domains with smooth boundaries the order of the second term ofN(λ) (under “billiard condition”) is one half of the dimension of the boundary. In the case of fractal boundaries the well-known Weyl-Berry hypothesis identifies it with one half of the Hausdorff dimension of ∂Ω, and the modified Weyl-Berry conjecture with one half of the Minkowski dimension of ∂Ω. We find the spectral asymptotics for three natural broad classes of fractal boundaries (cabbage type, bubble type and web type) and show that the Minkowski dimension gives the proper answer for cabbage type of boundaries (due to “one dimensional structure” of the cabbage type fractals), but the answers are principally different in the two other cases.     

Fractal basin boundaries

Basin boundaries for dynamical systems can be either smooth or fractal. This paper investigates fractal basin boundaries. One practical consequence of such boundaries is that they can lead to great difficulty in predicting to which attractor a system eventually goes. The structure of fractal basin boundaries can be classified as being either locally connected or locally disconnected. Examples and discussion of both types of structures are given, and it appears that fractal basin boundaries should be common in typical dynamical systems. Lyapunov numbers and the dimension for the measure generated by inverse orbits are also discussed.     

一种识别关联维数无标度区间的新方法

在计算关联维数过程中, 为了减少人为因素识别无标度区间带来的误差, 提出一种基于模拟退火遗传模糊C均值聚类识别无标度区间的新方法. 该方法根据无标度区间对应曲线的二阶导数在零附近波动的变化特征, 利用分类算法进行识别. 首先对双对数关联积分的离散数据进行二阶差分; 然后利用模拟退火遗传模糊C均值聚类方法对该数据进行分类, 选出在零附近波动的数据; 再剔除粗大误差保留有效数据; 最后进行统计分析识别出线性度最好的作为无标度区间. 应用新方法对两个著名的混沌系统Lorenz 和Henon 进行了仿真, 计算结果与理论值非常符合. 实验表明, 所提出的新方法与主观识别、K-means和2-means方法比较, 可以有效自动识别无标度区间, 减少误差, 计算结果更加精确.     

An Extended Correlation Dimension of Complex Networks

Fractal and self-similarity are important characteristics of complex networks. The correlation dimension is one of the measures implemented to characterize the fractal nature of unweighted structures, but it has not been extended to weighted networks. In this paper, the correlation dimension is extended to the weighted networks. The proposed method uses edge-weights accumulation to obtain scale distances. It can be used not only for weighted networks but also for unweighted networks. We selected six weighted networks, including two synthetic fractal networks and four real-world networks, to validate it. The results show that the proposed method was effective for the fractal scaling analysis of weighted complex networks. Meanwhile, this method was used to analyze the fractal properties of the Newman–Watts (NW) unweighted small-world networks. Compared with other fractal dimensions, the correlation dimension is more suitable for the quantitative analysis of small-world effects.     

Boundary Fractal Dimensions from Sections of a Particle Profile

The fractal dimension of particles is commonly evaluated from complete particle boundaries. In this work, a study has been made of the self-similar nature of complete and incomplete boundary profiles of a range of morphologically different copper powders. Boundary images were captured from SEM micrographs of particle boundaries at a range of magnifications up to nearly 14000X. An algorithm was developed to compute the fractal dimension of boundary segments. This algorithm was tested against the Koch Island fractal, and was found to give excellent estimates of the fractal dimension. For the particle system studied, the boundary fractal was found to be sensitive to magnification with appreciable drops in value at high magnification. This demonstrates that the particles studied did not have true fractal boundaries and the use of fractal theory to study particle surface roughness must be used with caution.     

Measuring capital market efficiency: Global and local correlations structure

We introduce a new measure for capital market efficiency. The measure takes into consideration the correlation structure of the returns (long-term and short-term memory) and local herding behavior (fractal dimension). The efficiency measure is taken as a distance from an ideal efficient market situation. The proposed methodology is applied to a portfolio of 41 stock indices. We find that the Japanese NIKKEI is the most efficient market. From a geographical point of view, the more efficient markets are dominated by the European stock indices and the less efficient markets cover mainly Latin America, Asia and Oceania. The inefficiency is mainly driven by a local herding, i.e. a low fractal dimension.     

The equality of fractal dimension and uncertainty dimension for certain dynamical systems

[MGOY] introduced the uncertainty dimension as a quantative measure for final state sensitivity in a system. In [MGOY] and [P] it was conjectured that the box-counting dimension equals the uncertainty dimension for basin boundaries in typical dynamical systems. In this paper our main result is that the box-counting dimension, the uncertainty dimension and the Hausdorff dimension are all equal for the basin boundaries of one and two dimensional systems, which are uniformly hyperbolic on their basin boundary. When the box-counting dimension of the basin boundary is large, that is, near the dimension of the phase space, this result implies that even a large decrease in the uncertainty of the position of the initial condition yields only a relatively small decrease in the uncertainty of which basin that initial point is in.Research in part supported by AFOSR and by the Department of Energy (Scientific Computing Staff Office of Energy Research)     

Monte Carlo simulations of catalytic CO oxidation on fractal surfaces of dimension between two and three

We present a method for generating fractal surfaces of dimension between two and three. By using the method, five fractal surfaces with dimension 2.262, 2.402, 2.524, 2.631, and 2.771 are created. For each of these surfaces, the reaction of carbon monoxide and oxygen is simulated by using a Monte Carlo method based on the ZGB model [Phys. Rev. Lett. 24 (1986) 2553]. The results show that the catalytic CO oxidation proceeds more efficiently on a surface with higher fractal dimension. It is also found that as the fractal dimension of the surface becomes higher, the first-order kinetic phase transition point (y₂) is shifted to a higher partial pressure of CO. This implies that poisoning of the catalyst surface due to CO segregation sets in at a higher CO partial pressure for surfaces with more complexity.     

基于图像分形相关位移测量新方法的研究

本文从分形理论出发提出了图像分形相关位移测量的新方法。较之20世纪80年代发展起来的以像素的灰度相关为基础的数字散斑相关法，分形相关法更充分地利用了散斑图的相关性特征，有关更深入的发展潜力。为了验证理论与方法的可靠性，进行了验证性实验并和数字散斑相关法的测试结果进行了比较。测试结果显示，测试精度至少可达0.06个像素。最后讨论了这一方法的发展前景。     

Fractal dimension of the grain boundaries in ceramics with nanodispersed additions

The fractal dimension of the grain boundaries in Al₂O3-MgO-SiO₂ corundum ceramic is measured. It is shown for the first time that a similarity exists between the aggregation of solid disperse particles and the grain formation in the ceramic and that this similarity can be used to reveal grain formation mechanisms. The fractal dimension of the grain boundaries is found to be 1.68 and 1.42 at sintering temperatures of 1200 and 1600°C, respectively. These values correspond to primary recrystallization and normal grain growth in the ceramic. A relationship between the fractal dimension of the grain boundaries and the sintering temperature of the corundum ceramic is obtained.     

Sound scattering by random fractal inhomogeneities in the ocean

Sound scattering by random volume inhomogeneities (fluctuations of the refraction index in a medium) with an arbitrary anisotropy is considered using the small perturbation method (Born’s approximation). Surfaces (boundaries) of the inhomogeneities are deemed to be fractal ones: the energy spectra of the refraction index fluctuations follow the power law with a nonintegral exponent. Formulas are obtained for the volume scattering coefficient. Frequency and angular dependences of the scattering coefficient and their relations to the fractal dimension of inhomogeneities with different kinds of anisotropy and different sizes (on the sound wavelength scale) are presented. The fractal dimension of the inhomogeneities is estimated.     

Infrared dim target detection based on fractal dimension and third-order characterization

We propose an improved algorithm based on fractal dimension and third-order characterization to detect dim target with cluttered background in an infrared （IR） image. We also illustrate the performance and efficiency comparisons between the presented algorithm and the traditional fractal detection method on real IR images. The experimental results show that the proposed algorithm is robust and efficient for IR dim target detection.     

A box-covering algorithm for fractal scaling in scale-free networks

A random sequential box-covering algorithm recently introduced to measure the fractal dimension in scale-free (SF) networks is investigated. The algorithm contains Monte Carlo sequential steps of choosing the position of the center of each box; thereby, vertices in preassigned boxes can divide subsequent boxes into more than one piece, but divided boxes are counted once. We find that such box-split allowance in the algorithm is a crucial ingredient necessary to obtain the fractal scaling for fractal networks; however, it is inessential for regular lattice and conventional fractal objects embedded in the Euclidean space. Next, the algorithm is viewed from the cluster-growing perspective that boxes are allowed to overlap; thereby, vertices can belong to more than one box. The number of distinct boxes a vertex belongs to is, then, distributed in a heterogeneous manner for SF fractal networks, while it is of Poisson-type for the conventional fractal objects.     

一种基于超复数系的数字全息图像生成方法

提出了一种基于超复数系的数字全息图像生成方法，通过定义超复数系，由此构成n维向量空间，然后采用分形迭代方法在n维向量空间中选择不同截面来绘制分形图像，因而能够在高维空间生成分形数字图像序列，并能够制作出具有动态效果和美感的激光全息图。由于分形数字全息图像生成具有较好的参数可控制性和不可逆转性，全盛的防伪标专难以仿制，在激光全息防伪领域有着良好的应用前景。     

应用分形理论探讨炸药的撞击感度

 根据分形几何的周长-面积公式及线性回归原理,对TATB等六种炸药样品的落锤试验的分幅图像进行了图像处理和分形维数计算。从中发现,越钝感的炸药样品,分形维数越大。因此,可以认为,用炸药样品的分形维数来鉴别撞击感度是基本可行的。     

Thermodynamically stable fractal-like domain structures in magnetic films

A fractal-like structure of the domain boundaries was revealed in “overcritical” uniaxial Permalloy magnetic films. The fractal dimension of domain boundaries at the film surfaces was determined as a function of the film thickness. It is shown that the phase transition between the two possible types of fractal-like structures is accompanied by a jump in fractal dimension.

     

Dynamics of coin tossing is predictable

The dynamics of the tossed coin can be described by deterministic equations of motion, but on the other hand it is commonly taken for granted that the toss of a coin is random. A realistic mechanical model of coin tossing is constructed to examine whether the initial states leading to heads or tails are distributed uniformly in phase space. We give arguments supporting the statement that the outcome of the coin tossing is fully determined by the initial conditions, i.e. no dynamical uncertainties due to the exponential divergence of initial conditions or fractal basin boundaries occur. We point out that although heads and tails boundaries in the initial condition space are smooth, the distance of a typical initial condition from a basin boundary is so small that practically any uncertainty in initial conditions can lead to the uncertainty of the results of tossing.     

A practical method to experimentally evaluate the Hausdorff dimension: an alternative phase-transition-based methodology

We introduce a methodology to estimate numerically the Hausdorff dimension of a geometric set. This practical method has been conceived as a subsequent tool of another context study, associated to our concern to distinguish between various fractal sets. Its conception is natural since it can be related to the original idea involved in the definitions of Hausdorff measure and Hausdorff dimension. It is based on the critical behavior of the measure spectrum functions of the set around its Hausdorff dimension value. We illustrate on several well-known examples, the ability of this method to accurately estimate the Hausdorff dimension. Also, we show how the transition property, exhibited by the quantities used as substitutes of the Hausdorff measure in the corresponding fractal dimension relationships, can be used to accurately estimate the fractal dimension. To show the potential of our method, we also report the results of Hausdorff dimension measurements on some typical examples, compared to a direct application of the scaling relation involved in the box-counting dimension definition.