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Power Law Behaviour in Complex Systems

A special issue of Entropy (ISSN 1099-4300). This special issue belongs to the section "Complexity".

Deadline for manuscript submissions: closed (30 April 2018) | Viewed by 45689

Special Issue Editors


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Guest Editor
Institute of Engineering, Department of Electrical Engineering, Polytechnic Institute of Porto, R. Dr. Roberto Frias, 4249-015 Porto, Portugal
Interests: nonlinear dynamics; complexity; fractional calculus; modeling; control; entropy; genomics
Special Issues, Collections and Topics in MDPI journals

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Guest Editor
Faculty of Engineering, University of Porto, Rua Dr. Roberto Frias, 4200–465 Porto, Portugal
Interests: complex systems modelling; automation and robotics; fractional order systems modelling and control; data analysis and visualization
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Complex systems (CS) are composed of a large number of autonomous, but non-independent, interacting entities. The resulting CS global dynamics is much richer than the one exhibited by the individual parts. Power law statistics is the most common description of complex dynamics. Since power law statistical distributions and fractional dynamics are connected, fractional order dynamics in often expected to occur in CS.

CS are ubiquitous in many areas, including financial markets, highway transportation networks, telecommunication networks, world and country economies, social networks, immunological systems, living organisms, computational systems and electrical and mechanical structures.

The Special Issue focuses on original and new research results on systems dynamics in science and engineering. Manuscripts in complex dynamical systems, nonlinearity, chaos and fractional dynamics in the thermodynamics or information processing perspectives are solicited. We welcome submissions addressing novel issues, as well as those on more specific topics illustrating the broad impact of entropy-based techniques in complexity, nonlinearity and fractionality.

Papers should fit the scope of the journal Entropy, namely they must include some content in Information Theory and/or Entropy in the text.

The main topics of interest include (but are not limited to):

- Complex dynamics
- Nonlinear dynamical systems
- Advanced control systems
- Entropy and Information Theory
- Fractional calculus and its applications
- Evolutionary computing
- Finance and economy dynamics
- Fractals and chaos
- Biological systems and bioinformatics
- Nonlinear waves and acoustics
- Image and signal processing
- Transportation systems
- Geosciences
- Astronomy and cosmology
- Nuclear physics

Prof. Dr. J. A. Tenreiro Machado
Prof. Dr. António M. Lopes
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Entropy is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • Power law
  • Dynamics
  • Complex systems
  • Fractional calculus
  • Entropy
  • Information theory

Published Papers (11 papers)

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Editorial

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3 pages, 157 KiB  
Editorial
Power Law Behaviour in Complex Systems
by António M. Lopes and J. A. Tenreiro Machado
Entropy 2018, 20(9), 671; https://doi.org/10.3390/e20090671 - 5 Sep 2018
Cited by 5 | Viewed by 2559
(This article belongs to the Special Issue Power Law Behaviour in Complex Systems)

Research

Jump to: Editorial

8 pages, 1019 KiB  
Article
Non-Linear Diffusion and Power Law Properties of Heterogeneous Systems: Application to Financial Time Series
by Miguel A. Fuentes
Entropy 2018, 20(9), 649; https://doi.org/10.3390/e20090649 - 30 Aug 2018
Cited by 6 | Viewed by 3403
Abstract
In this work, we show that it is possible to obtain important ubiquitous physical characteristics when an aggregation of many systems is taken into account. We discuss the possibility of obtaining not only an anomalous diffusion process, but also a Non-Linear diffusion equation, [...] Read more.
In this work, we show that it is possible to obtain important ubiquitous physical characteristics when an aggregation of many systems is taken into account. We discuss the possibility of obtaining not only an anomalous diffusion process, but also a Non-Linear diffusion equation, that leads to a probability distribution, when using a set of non-Markovian processes. This probability distribution shows a power law behavior in the structure of its tails. It also reflects the anomalous transport characteristics of the ensemble of particles. This ubiquitous behavior, with a power law in the diffusive transport and the structure of the probability distribution, is related to a fast fluctuating phenomenon presented in the noise parameter. We discuss all the previous results using a financial time series example. Full article
(This article belongs to the Special Issue Power Law Behaviour in Complex Systems)
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13 pages, 472 KiB  
Article
Diffusion on Middle-ξ Cantor Sets
by Alireza Khalili Golmankhaneh, Arran Fernandez, Ali Khalili Golmankhaneh and Dumitru Baleanu
Entropy 2018, 20(7), 504; https://doi.org/10.3390/e20070504 - 2 Jul 2018
Cited by 31 | Viewed by 4213
Abstract
In this paper, we study Cζ-calculus on generalized Cantor sets, which have self-similar properties and fractional dimensions that exceed their topological dimensions. Functions with fractal support are not differentiable or integrable in terms of standard calculus, so we must involve local [...] Read more.
In this paper, we study Cζ-calculus on generalized Cantor sets, which have self-similar properties and fractional dimensions that exceed their topological dimensions. Functions with fractal support are not differentiable or integrable in terms of standard calculus, so we must involve local fractional derivatives. We have generalized the Cζ-calculus on the generalized Cantor sets known as middle-ξ Cantor sets. We have suggested a calculus on the middle-ξ Cantor sets for different values of ξ with 0<ξ<1. Differential equations on the middle-ξ Cantor sets have been solved, and we have presented the results using illustrative examples. The conditions for super-, normal, and sub-diffusion on fractal sets are given. Full article
(This article belongs to the Special Issue Power Law Behaviour in Complex Systems)
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16 pages, 5098 KiB  
Article
Complexity Analysis of Global Temperature Time Series
by António M. Lopes and J. A. Tenreiro Machado
Entropy 2018, 20(6), 437; https://doi.org/10.3390/e20060437 - 5 Jun 2018
Cited by 8 | Viewed by 3712
Abstract
Climate has complex dynamics due to the plethora of phenomena underlying its evolution. These characteristics pose challenges to conducting solid quantitative analysis and reaching assertive conclusions. In this paper, the global temperature time series (TTS) is viewed as a manifestation of the climate [...] Read more.
Climate has complex dynamics due to the plethora of phenomena underlying its evolution. These characteristics pose challenges to conducting solid quantitative analysis and reaching assertive conclusions. In this paper, the global temperature time series (TTS) is viewed as a manifestation of the climate evolution, and its complexity is calculated by means of four different indices, namely the Lempel–Ziv complexity, sample entropy, signal harmonics power ratio, and fractal dimension. In the first phase, the monthly mean TTS is pre-processed by means of empirical mode decomposition, and the TTS trend is calculated. In the second phase, the complexity of the detrended signals is estimated. The four indices capture distinct features of the TTS dynamics in a 4-dim space. Hierarchical clustering is adopted for dimensional reduction and visualization in the 2-dim space. The results show that TTS complexity exhibits space-time variability, suggesting the presence of distinct climate forcing processes in both dimensions. Numerical examples with real-world data demonstrate the effectiveness of the approach. Full article
(This article belongs to the Special Issue Power Law Behaviour in Complex Systems)
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17 pages, 2343 KiB  
Article
Hierarchical Scaling in Systems of Natural Cities
by Yanguang Chen and Bin Jiang
Entropy 2018, 20(6), 432; https://doi.org/10.3390/e20060432 - 4 Jun 2018
Cited by 11 | Viewed by 4506
Abstract
Hierarchies can be modeled by a set of exponential functions, from which we can derive a set of power laws indicative of scaling. The solution to a scaling relation equation is always a power law. The scaling laws are followed by many natural [...] Read more.
Hierarchies can be modeled by a set of exponential functions, from which we can derive a set of power laws indicative of scaling. The solution to a scaling relation equation is always a power law. The scaling laws are followed by many natural and social phenomena such as cities, earthquakes, and rivers. This paper reveals the power law behaviors in systems of natural cities by reconstructing the urban hierarchy with cascade structure. Cities of the U.S.A., Britain, France, and Germany are taken as examples to perform empirical analyses. The hierarchical scaling relations can be well fitted to the data points within the scaling ranges of the number, size and area of the natural cities. The size-number and area-number scaling exponents are close to 1, and the size-area allometric scaling exponent is slightly less than 1. The results show that natural cities follow hierarchical scaling laws very well. The principle of entropy maximization of urban evolution is then employed to explain the hierarchical scaling laws, and differences entropy maximizing processes are used to interpret the scaling exponents. This study is helpful for scientists to understand the power law behavior in the development of cities and systems of cities. Full article
(This article belongs to the Special Issue Power Law Behaviour in Complex Systems)
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24 pages, 377 KiB  
Article
Criterion of Existence of Power-Law Memory for Economic Processes
by Vasily E. Tarasov and Valentina V. Tarasova
Entropy 2018, 20(6), 414; https://doi.org/10.3390/e20060414 - 29 May 2018
Cited by 23 | Viewed by 3614
Abstract
In this paper, we propose criteria for the existence of memory of power-law type (PLT) memory in economic processes. We give the criterion of existence of power-law long-range dependence in time by using the analogy with the concept of the long-range alpha-interaction. We [...] Read more.
In this paper, we propose criteria for the existence of memory of power-law type (PLT) memory in economic processes. We give the criterion of existence of power-law long-range dependence in time by using the analogy with the concept of the long-range alpha-interaction. We also suggest the criterion of existence of PLT memory for frequency domain by using the concept of non-integer dimensions. For an economic process, for which it is known that an endogenous variable depends on an exogenous variable, the proposed criteria make it possible to identify the presence of the PLT memory. The suggested criteria are illustrated in various examples. The use of the proposed criteria allows apply the fractional calculus to construct dynamic models of economic processes. These criteria can be also used to identify the linear integro-differential operators that can be considered as fractional derivatives and integrals of non-integer orders. Full article
(This article belongs to the Special Issue Power Law Behaviour in Complex Systems)
11 pages, 378 KiB  
Article
Time-Fractional Diffusion with Mass Absorption in a Half-Line Domain due to Boundary Value of Concentration Varying Harmonically in Time
by Yuriy Povstenko and Tamara Kyrylych
Entropy 2018, 20(5), 346; https://doi.org/10.3390/e20050346 - 6 May 2018
Cited by 9 | Viewed by 3046
Abstract
The time-fractional diffusion equation with mass absorption is studied in a half-line domain under the Dirichlet boundary condition varying harmonically in time. The Caputo derivative is employed. The solution is obtained using the Laplace transform with respect to time and the sin-Fourier transform [...] Read more.
The time-fractional diffusion equation with mass absorption is studied in a half-line domain under the Dirichlet boundary condition varying harmonically in time. The Caputo derivative is employed. The solution is obtained using the Laplace transform with respect to time and the sin-Fourier transform with respect to the spatial coordinate. The results of numerical calculations are illustrated graphically. Full article
(This article belongs to the Special Issue Power Law Behaviour in Complex Systems)
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20 pages, 1710 KiB  
Article
Finite Difference Method for Time-Space Fractional Advection–Diffusion Equations with Riesz Derivative
by Sadia Arshad, Dumitru Baleanu, Jianfei Huang, Maysaa Mohamed Al Qurashi, Yifa Tang and Yue Zhao
Entropy 2018, 20(5), 321; https://doi.org/10.3390/e20050321 - 26 Apr 2018
Cited by 28 | Viewed by 5053
Abstract
In this article, a numerical scheme is formulated and analysed to solve the time-space fractional advection–diffusion equation, where the Riesz derivative and the Caputo derivative are considered in spatial and temporal directions, respectively. The Riesz space derivative is approximated by the second-order fractional [...] Read more.
In this article, a numerical scheme is formulated and analysed to solve the time-space fractional advection–diffusion equation, where the Riesz derivative and the Caputo derivative are considered in spatial and temporal directions, respectively. The Riesz space derivative is approximated by the second-order fractional weighted and shifted Grünwald–Letnikov formula. Based on the equivalence between the fractional differential equation and the integral equation, we have transformed the fractional differential equation into an equivalent integral equation. Then, the integral is approximated by the trapezoidal formula. Further, the stability and convergence analysis are discussed rigorously. The resulting scheme is formally proved with the second order accuracy both in space and time. Numerical experiments are also presented to verify the theoretical analysis. Full article
(This article belongs to the Special Issue Power Law Behaviour in Complex Systems)
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26 pages, 6106 KiB  
Article
The Power Law Characteristics of Stock Price Jump Intervals: An Empirical and Computational Experimental Study
by Hongduo Cao, Hui Ouyang, Ying Li, Xiaobin Li and Ye Chen
Entropy 2018, 20(4), 304; https://doi.org/10.3390/e20040304 - 21 Apr 2018
Cited by 2 | Viewed by 4346
Abstract
For the first time, the power law characteristics of stock price jump intervals have been empirically found generally in stock markets. The classical jump-diffusion model is described as the jump-diffusion model with power law (JDMPL). An artificial stock market (ASM) is designed in [...] Read more.
For the first time, the power law characteristics of stock price jump intervals have been empirically found generally in stock markets. The classical jump-diffusion model is described as the jump-diffusion model with power law (JDMPL). An artificial stock market (ASM) is designed in which an agent’s investment strategies, risk appetite, learning ability, adaptability, and dynamic changes are considered to create a dynamically changing environment. An analysis of these data packets from the ASM simulation indicates that, with the learning mechanism, the ASM reflects the kurtosis, fat-tailed distribution characteristics commonly observed in real markets. Data packets obtained from simulating the ASM for 5010 periods are incorporated into a regression analysis. Analysis results indicate that the JDMPL effectively characterizes the stock price jumps in the market. The results also support the hypothesis that the time interval of stock price jumps is consistent with the power law and indicate that the diversity and dynamic changes of agents’ investment strategies are the reasons for the discontinuity in the changes of stock prices. Full article
(This article belongs to the Special Issue Power Law Behaviour in Complex Systems)
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15 pages, 409 KiB  
Article
A Simple and Adaptive Dispersion Regression Model for Count Data
by Hadeel S. Klakattawi, Veronica Vinciotti and Keming Yu
Entropy 2018, 20(2), 142; https://doi.org/10.3390/e20020142 - 22 Feb 2018
Cited by 28 | Viewed by 5911
Abstract
Regression for count data is widely performed by models such as Poisson, negative binomial (NB) and zero-inflated regression. A challenge often faced by practitioners is the selection of the right model to take into account dispersion, which typically occurs in count datasets. It [...] Read more.
Regression for count data is widely performed by models such as Poisson, negative binomial (NB) and zero-inflated regression. A challenge often faced by practitioners is the selection of the right model to take into account dispersion, which typically occurs in count datasets. It is highly desirable to have a unified model that can automatically adapt to the underlying dispersion and that can be easily implemented in practice. In this paper, a discrete Weibull regression model is shown to be able to adapt in a simple way to different types of dispersions relative to Poisson regression: overdispersion, underdispersion and covariate-specific dispersion. Maximum likelihood can be used for efficient parameter estimation. The description of the model, parameter inference and model diagnostics is accompanied by simulated and real data analyses. Full article
(This article belongs to the Special Issue Power Law Behaviour in Complex Systems)
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26 pages, 341 KiB  
Article
Is Natural Language a Perigraphic Process? The Theorem about Facts and Words Revisited
by Łukasz Dębowski
Entropy 2018, 20(2), 85; https://doi.org/10.3390/e20020085 - 26 Jan 2018
Cited by 15 | Viewed by 4280
Abstract
As we discuss, a stationary stochastic process is nonergodic when a random persistent topic can be detected in the infinite random text sampled from the process, whereas we call the process strongly nonergodic when an infinite sequence of independent random bits, called probabilistic [...] Read more.
As we discuss, a stationary stochastic process is nonergodic when a random persistent topic can be detected in the infinite random text sampled from the process, whereas we call the process strongly nonergodic when an infinite sequence of independent random bits, called probabilistic facts, is needed to describe this topic completely. Replacing probabilistic facts with an algorithmically random sequence of bits, called algorithmic facts, we adapt this property back to ergodic processes. Subsequently, we call a process perigraphic if the number of algorithmic facts which can be inferred from a finite text sampled from the process grows like a power of the text length. We present a simple example of such a process. Moreover, we demonstrate an assertion which we call the theorem about facts and words. This proposition states that the number of probabilistic or algorithmic facts which can be inferred from a text drawn from a process must be roughly smaller than the number of distinct word-like strings detected in this text by means of the Prediction by Partial Matching (PPM) compression algorithm. We also observe that the number of the word-like strings for a sample of plays by Shakespeare follows an empirical stepwise power law, in a stark contrast to Markov processes. Hence, we suppose that natural language considered as a process is not only non-Markov but also perigraphic. Full article
(This article belongs to the Special Issue Power Law Behaviour in Complex Systems)
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