We report on a quantitative analysis of relationships between the number of homicides, population size and ten other urban metrics. By using data from Brazilian cities, we show that well-defined ...average scaling laws with the population size emerge when investigating the relations between population and number of homicides as well as population and urban metrics. We also show that the fluctuations around the scaling laws are log-normally distributed, which enabled us to model these scaling laws by a stochastic-like equation driven by a multiplicative and log-normally distributed noise. Because of the scaling laws, we argue that it is better to employ logarithms in order to describe the number of homicides in function of the urban metrics via regression analysis. In addition to the regression analysis, we propose an approach to correlate crime and urban metrics via the evaluation of the distance between the actual value of the number of homicides (as well as the value of the urban metrics) and the value that is expected by the scaling law with the population size. This approach has proved to be robust and useful for unveiling relationships/behaviors that were not properly carried out by the regression analysis, such as Formula: see text the non-explanatory potential of the elderly population when the number of homicides is much above or much below the scaling law, Formula: see text the fact that unemployment has explanatory potential only when the number of homicides is considerably larger than the expected by the power law, and Formula: see text a gender difference in number of homicides, where cities with female population below the scaling law are characterized by a number of homicides above the power law.
Complexity measures are essential to understand complex systems and there are numerous definitions to analyze one-dimensional data. However, extensions of these approaches to two or ...higher-dimensional data, such as images, are much less common. Here, we reduce this gap by applying the ideas of the permutation entropy combined with a relative entropic index. We build up a numerical procedure that can be easily implemented to evaluate the complexity of two or higher-dimensional patterns. We work out this method in different scenarios where numerical experiments and empirical data were taken into account. Specifically, we have applied the method to Formula: see text fractal landscapes generated numerically where we compare our measures with the Hurst exponent; Formula: see text liquid crystal textures where nematic-isotropic-nematic phase transitions were properly identified; Formula: see text 12 characteristic textures of liquid crystals where the different values show that the method can distinguish different phases; Formula: see text and Ising surfaces where our method identified the critical temperature and also proved to be stable.
A quantum motion of a particle along the x-direction constrained at y=0 in presence of the Dirac δ-potential is considered. The Green’s function approach is applied for solving the corresponding ...two-dimensional Schrödinger equation. The exact results for the Green’s function are presented in terms of the Fox H-function, and a generalized integral operator which contains the Mittag-Leffler function in the kernel.
More than a half of world population is now living in cities and this number is expected to be two-thirds by 2050. Fostered by the relevancy of a scientific characterization of cities and for the ...availability of an unprecedented amount of data, academics have recently immersed in this topic and one of the most striking and universal finding was the discovery of robust allometric scaling laws between several urban indicators and the population size. Despite that, most governmental reports and several academic works still ignore these nonlinearities by often analyzing the raw or the per capita value of urban indicators, a practice that actually makes the urban metrics biased towards small or large cities depending on whether we have super or sublinear allometries. By following the ideas of Bettencourt et al. PLoS ONE 5 (2010) e13541, we account for this bias by evaluating the difference between the actual value of an urban indicator and the value expected by the allometry with the population size. We show that this scale-adjusted metric provides a more appropriate/informative summary of the evolution of urban indicators and reveals patterns that do not appear in the evolution of per capita values of indicators obtained from Brazilian cities. We also show that these scale-adjusted metrics are strongly correlated with their past values by a linear correspondence and that they also display crosscorrelations among themselves. Simple linear models account for 31%-97% of the observed variance in data and correctly reproduce the average of the scale-adjusted metric when grouping the cities in above and below the allometric laws. We further employ these models to forecast future values of urban indicators and, by visualizing the predicted changes, we verify the emergence of spatial clusters characterized by regions of the Brazilian territory where we expect an increase or a decrease in the values of urban indicators.
We study the entropy production in a fractal system composed of two subsystems, each of which is subjected to an external force. This is achieved by using the H-theorem on the nonlinear Fokker-Planck ...equations (NFEs) characterizing the diffusing dynamics of each subsystem. In particular, we write a general NFE in terms of Hausdorff derivatives to take into account the metric of each system. We have also investigated some solutions from the analytical and numerical point of view. We demonstrate that each subsystem affects the total entropy and how the diffusive process is anomalous when the fractal nature of the system is considered.
We investigate the dynamics of a system composed of two different subsystems when subjected to different nonlinear Fokker–Planck equations by considering the H–theorem. We use the H–theorem to obtain ...the conditions required to establish a suitable dependence for the system’s interaction that agrees with the thermodynamics law when the nonlinearity in these equations is the same. In this framework, we also consider different dynamical aspects of each subsystem and investigate a possible expression for the entropy of the composite system.
We investigate the H-theorem for a class of generalized kinetic equations with fractional time-derivative, hyperbolic term, and nonlinear diffusion. When the H-theorem is satisfied, we demonstrate ...that different entropic forms may emerge due to the equation’s nonlinearity. We obtain the entropy production related to these entropies and show that its form remains invariant. Furthermore, we investigate some behaviors for these equations from both numerical and analytical perspectives, showing a large class of behaviors connected with anomalous diffusion and their effects on entropy.
The generalized diffusion equations with fractional order derivatives have shown be quite efficient to describe the diffusion in complex systems, with the advantage of producing exact expressions for ...the underlying diffusive properties. Recently, researchers have proposed different fractional-time operators (namely: the Caputo-Fabrizio and Atangana-Baleanu) which, differently from the well-known Riemann-Liouville operator, are defined by non-singular memory kernels. Here we proposed to use these new operators to generalize the usual diffusion equation. By analyzing the corresponding fractional diffusion equations within the continuous time random walk framework, we obtained waiting time distributions characterized by exponential, stretched exponential, and power-law functions, as well as a crossover between two behaviors. For the mean square displacement, we found crossovers between usual and confined diffusion, and between usual and sub-diffusion. We obtained the exact expressions for the probability distributions, where non-Gaussian and stationary distributions emerged. This former feature is remarkable because the fractional diffusion equation is solved without external forces and subjected to the free diffusion boundary conditions. We have further shown that these new fractional diffusion equations are related to diffusive processes with stochastic resetting, and to fractional diffusion equations with derivatives of distributed order. Thus, our results suggest that these new operators may be a simple and efficient way for incorporating different structural aspects into the system, opening new possibilities for modeling and investigating anomalous diffusive processes.
Abstract
Recent research has shown that criminal networks have complex organizational structures, but whether this can be used to predict static and dynamic properties of criminal networks remains ...little explored. Here, by combining graph representation learning and machine learning methods, we show that structural properties of political corruption, police intelligence, and money laundering networks can be used to recover missing criminal partnerships, distinguish among different types of criminal and legal associations, as well as predict the total amount of money exchanged among criminal agents, all with outstanding accuracy. We also show that our approach can anticipate future criminal associations during the dynamic growth of corruption networks with significant accuracy. Thus, similar to evidence found at crime scenes, we conclude that structural patterns of criminal networks carry crucial information about illegal activities, which allows machine learning methods to predict missing information and even anticipate future criminal behavior.
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