China
Africa
Explore chapters and articles related to this topic
Chaos in Space
Published in Pier Luigi Gentili, Untangling Complex Systems, 2018
An effective strategy to recognize power laws is to build log-log plots of experimental data. In fact, a true power law appears as a straight line in a double logarithmic axis plot (see Figure 11.21). Most of the natural, social, and economic phenomena, cited earlier, when analyzed in a sufficiently wide range of possible values of their variables, may exhibit a limited linear regime followed by a significant curvature. Such trend is well-fitted by a stretched exponential function of the type [] y=Ae[−(xx0)c]
Fractals
Published in Nicholas Stergiou, Nonlinear Analysis for Human Movement Variability, 2018
Note that Equation 7.3 has the same format as the equation of a line (y=b+ax), where log y is y, log A is b, n is a, and log x is x. Therefore, when we plot our log–log plot, we can easily identify the slope of the line (Figure 7.10). This information is critical in understanding how we can measure the fractal properties of a human movement time series. Investigating whether data appear as a straight line on a log–log scale can be instructive in detecting possible power laws. In the next section, we will explore what kind of data need to be plotted on a log–log plot, and what power laws mean for human movement.
General power laws of the causalities in the causal Bayesian networks
Published in International Journal of General Systems, 2023
Boyuan Li, Xiaoyang Li, Zhaoxing Tian, Xia Lu, Rui Kang
In general, networks are utilized to model the factors and interactions in complex systems, and the critical phenomena in complex systems are manifested as the power laws of some network properties. Current researches mainly focus on the power laws of network geometric properties and corresponding phase transitions, i.e. geometric phase transitions. And the network geometric properties mainly include: (1). The connectivity of networks. When the probability that two nodes in a network are connected is greater than some threshold, the connectivity of the whole network is significantly improved, and this is known as the percolation transitions (Cohen, Ben-Avraham, and Havlin 2002); (2). The denseness of networks. Studies showed that the degrees (Albert, Jeong, and Barabási 1999), clustering coefficients (Dunne, Williams, and Martinez 2002), and the “mass” related to fractal (Song, Havlin, and Makse 2005; Daqing et al. 2011) follow the power laws in some systems. And such power laws indicate the phase transitions of the networks’ denseness; (3). The importance of the nodes in a network. The phase transitions of this property are mainly reflected in the power laws of the nodes’ centralities (Barthelemy 2004; Goh, Kahng, and Kim 2001; Jain and Sinha 2023).
The fractal or scaling perspective on progressively generated intra-urban clusters from street junctions
Published in International Journal of Digital Earth, 2023
Biao He, Renzhong Guo, Minmin Li, Ying Jing, Zhigang Zhao, Wei Zhu, Chen Zhang, Chengyue Zhang, Ding Ma
The common range of exponent value is between 1 and 3. As a power law distribution demonstrates a substantial imbalance or heterogeneity, we can say that the more heterogenous the data, the larger the exponent will be. Furthermore, to evaluate the adherence of empirical data to an ideal power-law model fitted with and values, a modified Kolmogorov–Smirnov test suggested by Clauset, Shalizi, and Newman (2009) can be utilized. Simply put, this test involves generating numerous synthetic datasets that conform to a perfect power-law above , while retaining the same non-power-law distribution as the original dataset below . By comparing the synthetic datasets to the fitted model and calculating the maximum difference between them, the goodness-of-fit index p-value can be obtained. The p-value ranges from 0 to 1 and is determined by calculating the ratio of the number of synthetic datasets that display a greater maximum difference than the original dataset to the total number of synthetic datasets. A p-value of 0 signifies the rejection of the hypothesis that the dataset follows a power-law distribution. This procedure is repeated 1000 times to ensure accuracy. In this study, the baseline of p-value is set as no smaller than 0.05. This means that among 1000 synthetic datasets, at least 50 should have a weaker power-law distribution than the original dataset, thereby suggesting the original data obeys a power law distribution.
Rainfall flood hazard at nuclear power plants in India
Published in Georisk: Assessment and Management of Risk for Engineered Systems and Geohazards, 2018
Rainfall data are generally fitted with Gumbel distribution, however in the current analysis, it has been seen that it is more suited for fitting inland regions. Gumbel distribution gives slightly less predicted rainfall than other fitted models. The exponential distribution gives higher predicted rainfall than Gumbel model but lower than that of the power law. The Power law approach combines maximum-likelihood fitting methods with goodness-of-fit tests based on the Kolmogorov–Smirnov statistic (Clauset, Shalizi, and Newman 2009). The lower bound for empirical data which follows power law is approximately 110–120 mm for all site-specific data set. In this analysis, 100 mm is chosen as lower bound for all data set.