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.     

Fractal Analysis of Mobile Social Networks

Fractal and self similarity of complex networks have attracted much attention in recent years.The fractal dimension is a useful method to describe the fractal property of networks.However,the fractal features of mobile social networks(MSNs) are inadequately investigated.In this work,a box-covering method based on the ratio of excluded mass to closeness centrality is presented to investigate the fractal feature of MSNs.Using this method,we find that some MSNs are fractal at different time intervals.Our simulation results indicate that the proposed method is available for analyzing the fractal property of MSNs.     

Weighted Fractal Networks

In this paper we define a new class of weighted complex networks sharing several properties with fractal sets, and whose topology can be completely analytically characterized in terms of the involved parameters and of the fractal dimension. General networks with fractal or hierarchical structures can be set in the proposed framework that moreover could be used to provide some answers to the widespread emergence of fractal structures in nature.     

断裂表面自相似与自仿射复杂分形结构的研究

 按照分形理论研究了具有自相似和自仿射分形结构的复杂表面,并研究了其在Ⅰ型+Ⅱ型和Ⅰ型+Ⅲ型混合载荷下形成的断裂表面的应用。结果指出：分维D与粗糙度指数H之间的关系（D+H=2或D×H=1）只是粗略的近似。用分维D或粗糙度指数H测量断裂表面,在双对数图上得到的是实验曲线而不是直线并不说明没有分形结构,很可能是自相似和自仿射结构的混合。     

A review of fractality and self-similarity in complex networks

We review recent findings of self-similarity in complex networks. Using the box-covering technique, it was shown that many networks present a fractal behavior, which is seemingly in contrast to their small-world property. Moreover, even non-fractal networks have been shown to present a self-similar picture under renormalization of the length scale. These results have an important effect in our understanding of the evolution and behavior of such systems. A large number of network properties can now be described through a set of simple scaling exponents, in analogy with traditional fractal theory.     

花簇分形无标度网络中节点影响力的区分度

大量研究表明分形尺度特性广泛存在于真实复杂系统中, 且分形结构显著影响网络上的传播动力学行为. 虽然复杂网络的节点传播影响力吸引了越来越多学者的关注, 但依旧缺乏针对分形网络结构的节点影响力的系统研究. 鉴于此, 本文基于花簇分形网络模型, 研究了分形无标度结构上的节点传播影响力. 首先, 对比了不同分形维数下的节点影响力, 结果表明, 当分形维数很小时, 节点影响力的区分度几乎不随节点度变化, 很难区分不同节点的传播影响力, 而随着分形维数的增大, 从全局和局域角度都能很容易识别网络中的超级传播源. 其次, 通过对原分形网络进行不同程度的随机重连来分析网络噪声对节点影响力区分度的影响, 发现在低维分形网络上, 加入网络噪声之后能够容易区分不同节点的影响力, 而在无穷维超分形网络中, 加入网络噪声之后能够区分中间度节点的影响力, 但从全局和局域角度都很难识别中心节点的影响力. 所得结论进一步补充、深化了基于花簇分形网络的节点影响力研究, 研究结果对实际病毒传播的预警控制提供了一定的理论借鉴.     

Approach to an irregular time series on the basis of the fractal theory

We present a technique to measure the fractal dimension of the set of points (t, f(t)) forming the graph of a function f defined on the unit interval. First we apply it to a fractional Brownian function [1] which has a property of self-similarity for all scales, and we can get the stable and precise fractal dimension. This technique is also applied to the observational data of natural phenomena. It does not show self-similarity all over the scale but has a different self-similarity across the characteristic time scale. The present method gives us a stable characteristic time scale as well as the fractal dimension.     

分形理论在光谱识别中的应用

分形理论是研究一类不规则、混乱复杂,但其局部和整体具有相似性体系的科学。分形维数是分形理论中用于描述对象的不规则度和自相似性的基本度量。文章以符合朗伯-比尔定律的光谱信号为研究对象,在概述分形几何基本原理的基础上,提出了以分形维数作为光谱识别特征的方法,运用相空间重构得出了光谱信号的分形维数,通过对光谱信号的分形维数进行比较,达到识别不同光谱的目的,最后举例对该方法进行了说明。     

Role of fractal dimension in random walks on scale-free networks

Fractal dimension is central to understanding dynamical processes occurring on networks; however, the relation between fractal dimension and random walks on fractal scale-free networks has been rarely addressed, despite the fact that such networks are ubiquitous in real-life world. In this paper, we study the trapping problem on two families of networks. The first is deterministic, often called (x,y)-flowers; the other is random, which is a combination of (1,3)-flower and (2,4)-flower and thus called hybrid networks. The two network families display rich behavior as observed in various real systems, as well as some unique topological properties not shared by other networks. We derive analytically the average trapping time for random walks on both the (x,y)-flowers and the hybrid networks with an immobile trap positioned at an initial node, i.e., a hub node with the highest degree in the networks. Based on these analytical formulae, we show how the average trapping time scales with the network size. Comparing the obtained results, we further uncover that fractal dimension plays a decisive role in the behavior of average trapping time on fractal scale-free networks, i.e., the average trapping time decreases with an increasing fractal dimension.     

Embedding complex networks in a low dimensional Euclidean space based on vertex dissimilarities

We propose a method for representing vertices of a complex network as points in a Euclidean space of an appropriate dimension. To this end, we first adopt two widely used quantities as the measures for the dissimilarity between vertices. The dissimilarity is then transformed into its corresponding distance in a Euclidean space via the non-metric multidimensional scaling. We applied the proposed method to real-world as well as models of complex networks. We empirically found that real-world complex networks were embedded in a Euclidean space of relatively lower dimensions and the configuration of vertices in the space was mostly characterized by the self-similarity of a multifractal. In contrast, by applying the same scheme to the network models, we found that, in general, higher dimensions were needed to embed the networks into a Euclidean space and the embedding results usually did not exhibit the self-similar property. From the analysis, we learn that the proposed method serves a way not only to visualize the complex networks in a Euclidean space but to characterize the complex networks in a different manner from conventional ways.     

Fractal analysis on human dynamics of library loans

In this paper, the fractal characteristic of human behaviors is investigated from the perspective of time series constructed with the amount of library loans. The values of the Hurst exponent and length of non-periodic cycle calculated through rescaled range analysis indicate that the time series of human behaviors and their sub-series are fractal with self-similarity and long-range dependence. Then the time series are converted into complex networks by the visibility algorithm. The topological properties of the networks such as scale-free property and small-world effect imply that there is a close relationship among the numbers of repetitious behaviors performed by people during certain periods of time. Our work implies that there is intrinsic regularity in the human collective repetitious behaviors. The conclusions may be helpful to develop some new approaches to investigate the fractal feature and mechanism of human dynamics, and provide some references for the management and forecast of human collective behaviors.     

时频域分形维数分析的光谱信号重叠峰解析算法

由于光谱谱线存在自然展宽、多普勒展宽、碰撞展宽等,使混合气体中多种成分的吸收光谱信号出现相邻谱峰重叠现象,给混合气体组成成分的定性或定量检测带来较大的困难。现有的方法在获取先验知识、处理精度、运算效率等方面存在不足。提出基于时频域分形维数分析的光谱信号重叠峰解析算法,结合小波的多尺度观测能力和分形的自相似度的度量能力,识别、定位和解析光谱信号中的重叠峰。首先利用小波对具有重叠谱峰的光谱信号进行光谱频率域和尺度域的分析,然后对该时频域的光谱信号在同一光谱频率下的多尺度数据进行自相似性度量和分形计算。逐频率计算后得到光谱信号在频率域的分形维数曲线。该曲线体现了光谱信号在不同尺度的自相似性,其极值位置与光谱信号的各独立峰的位置具有相关性。依据此特性,结合分形曲线的特征参数,最后利用神经网络解析出对应混合气体成分的混叠在一起的各个独立谱峰。该方法利用小波的多分辨率特性,对信号进行不同尺度的精细度量。分形模型则提高了系统解析复杂信号的能力,对重叠程度高的多谱峰重叠信号也有很强的处理能力。借助人工神经网络,实现了整个算法的自动测量。通过实验结果分析,验证了算法的有效性,并讨论影响算法效果的主要因素。     

分形信号算法的CASI高光谱数据岩性特征提取

高光谱数据具有图谱合一和数据量大的特点,数据降维是主要的研究方向。波段选择和特征提取是目前高光谱降维的主要方法,就高光谱数据图像岩性特征提取的方法进行了试验和探讨。基于高光谱影像的自相似特征, 探索了分形信号算法在CASI高光谱数据岩性特征提取上的应用研究。以CASI高光谱影像数据为研究对象, 将基于地毯的方法进行修正后用于计算高光谱影像中每一像元的分形信号值。试验结果表明, 与其他分类算法相比分形信号算法增强高光谱图像的影像特征从另一个侧面更细致的描述了不同光谱的可区分性。分形信号影像在一定程度上可以更好地突出基岩裸露地区岩性特征, 从而可以实现影像地表岩性特征提取的目的。原始光谱曲线自身形态特征、初始尺度的选择以及迭代步长等对分形信号和分形特征尺度均有影响。目前,光谱曲线的分形信号特征研究还不多,对其物理意义和定量分析尚需要深入研究。     

The fractal measurement of experimental images of supersonic turbulent mixing layer

Flow visualization of supersonic mixing layer has been studied based on the high spatiotemporal resolution Nano-based Planar Laser Scattering(NPLS) method in SML-1 wind tunnel. The corresponding images distinctly reproduced the flow structure of laminar,transitional and turbulent region,with which the fractal measurement can be implemented. Two methods of measuring fractal dimension were introduced and compared. The fractal dimension of the transitional region and the fully developing turbulence region of supersonic mixing layer were measured based on the box-counting method. In the transitional region,the fractal dimension will increase with turbulent intensity. In the fully developing turbulent region,the fractal dimension will not vary apparently for different flow structures,which em-bodies the self-similarity of supersonic turbulence.     

气体吸附法测定二氧化硅干凝胶的分形维数

提出了一种方便、科学有效的利用气体吸附法测定二氧化硅干凝胶等多孔材料分形维数（表面分形维数和孔分布分形维数）的方法，不需要进行一系列的吸附/脱附实验，只需要利用单一气体的一次吸附/脱附实验得出的样品孔分布、比表面数据，与不同的标尺进行关联，即可同时获得表面分形维数和孔分布分形维数.通过误差分析和校正，保证了结果的可靠性.用上述方法测定了二氧化硅干凝胶的分形维数，以FHH法和SAXS法对所得结果进行了比较和验证，并对吸附/脱附过程所得结果的差异进行了初步分析. 关键词： 分形维数 气体吸附 二氧化硅 干凝胶     

基于区域分维和非采样Contourlet变换的图像融合算法

针对源图像有用信息的提取,提出了基于区域分维和非下采样Contourlet变换相结合的红外与可见光图像融合算法.将图像的区域属性、区域大小、边缘强度以及纹理显著程度等特点用图像不同尺度上的区域分维进行描述,对于非下采样Contourlet变换低频系数,根据源图像不同尺度上的区域分维进行基于系数选择的融合.针对带通子带系数设计了系数局部匹配度算子,依据匹配度不同采用加权和系数选取相结合的融合规则.与其他常规融合方法进行比较,该算法可有效实现红外与可见光图像的融合.     

分形物体转动惯量的计算

由分形物体的自相似性、转动惯量的量纲和平行轴定理,分别计算并得到分形三角形、分形正方体、分形四面体和科赫雪花的转动惯量.     

随机分布烟尘团簇粒子辐射特性研究

基于分形理论,采用蒙特卡罗方法对随机分布的烟尘团簇粒子结构进行了仿真模拟,利用离散偶极子近似(discrete dipole approximation, DDA)方法研究了随机分布的烟尘团簇粒子的辐射特性,分析讨论了分形维数、原始微粒粒径和数量以及复折射率对随机分布烟尘团簇粒子辐射特性的影响.研究表明,在给定分形维数的情况下,烟尘团簇粒子的辐射特性取决于原始微粒粒径、数量及复折射率;原始微粒较小的团簇粒子,当分形维数较小时,吸收截面变化不明显,但当分形维数大于2时,吸收截面骤然增大,然而,对于具有比较大的原始微粒粒径、数量及复折射率的烟尘团簇粒子,吸收截面随着分形维数的增大而单调递减;随着分形维数的增大,团簇粒子的散射截面、消光截面及单次散射反照率均单调递增;从整体上来讲,团簇粒子的辐射特性与等效球形粒子的辐射特性存在着比较大的差别,并且这种差别随着分形维数的增大而减小.该工作对研究气溶胶粒子的辐射及气候效应具有重要的科学价值. 关键词： 烟尘团簇粒子 辐射特性 离散偶极子近似方法     

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.     

FRACTAL PATTERN GROWTH OF METAL ATOM CLUSTERS IN ION IMPLANTED POLYMERS

The fractal and multi-fractal patterns of metal atoms are observed in the surface layer and cross section of a metal ion implanted polymer using TEM and SEM for the first time. The surface structure in the metal ion implanted polyethylene terephthalane (PET) is the random fractal. Certain average quantities of the random geometric patterns contain self-similarity. Some growth origins appeared in the fractal pattern which has a dimension of 1.67. The network structure of the fractal patterns is formed in cross section, having a fractal dimension of 1.87. So it can be seen that the fractal pattern is three-dimensional space fractal. We also find the collision cascade fractal in the cross section of implanted nylon, which is similar to the collision cascade pattern in transverse view calculated by the TRIM computer program. Finally, the mechanism for the formation and growth of the fractal patterns during ion implantation is discussed.