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Results for nonlinear diffusion equations with stochastic resetting

Ervin K. Lenzi, Rafael S. Zola, Michely P. Rosseto, Renio S. Mendes, Haroldo V. Ribeiro, Luciano R. da Silva, Luiz R. Evangelista
Entropy 25, 1647 (2023)
Anomalous Diffusion

We investigate a nonlinear diffusion process in which particles stochastically reset to their initial positions at a constant rate. The nonlinear diffusion process is modeled using the porous media equation and its extensions, which are nonlinear diffusion equations. We use analytical and numerical calculations to obtain and interpret the probability distribution of the position of the particles and the mean square displacement. These results are further compared and shown to agree with the results of numerical simulations. Our findings show that a system of this kind exhibits non-Gaussian distributions, transient anomalous diffusion (subdiffusion and superdiffusion), and stationary states that simultaneously depend on the nonlinearity and resetting rate.


Universal productivity patterns in research careers

Andre S. Sunahara, Matjaz Perc, Haroldo V. Ribeiro
Physical Review Research 5, 043203 (2023)
Science of Science

A common expectation is that career productivity peaks rather early and then gradually declines with seniority. But whether this holds true is still an open question. Here we investigate the productivity trajectories of almost 8,500 scientists from over fifty disciplines using methods from time series analysis, dimensionality reduction, and network science, showing that there exist six universal productivity patterns in research. Based on clusters of productivity trajectories and network representations where researchers with similar productivity patterns are connected, we identify constant, u-shaped, decreasing, periodic-like, increasing, and canonical productivity patterns, with the latter two describing almost three-fourths of researchers. In fact, we find that canonical curves are the most prevalent, but contrary to expectations, productivity peaks occur much more frequently around mid-career rather than early. These results outline the boundaries of possible career paths in science and caution against the adoption of stereotypes in tenure and funding decisions.


Interplay between particle trapping and heterogeneity in anomalous diffusion

Haroldo V. Ribeiro, Angel A. Tateishi, Ervin K. Lenzi, Richard L. Magin, Matjaz Perc
Communications Physics 6, 244 (2023)
Anomalous DiffusionComb Model

Heterogeneous media diffusion is often described using position-dependent diffusion coefficients and estimated indirectly through mean squared displacement in experiments. This approach may overlook other mechanisms and their interaction with position-dependent diffusion, potentially leading to erroneous conclusions. Here, we introduce a hybrid diffusion model that merges a position-dependent diffusion coefficient with the trapping mechanism of the comb model. We derive exact solutions for position distributions and mean squared displacements, validated through simulations of Langevin equations. Our model shows that the trapping mechanism attenuates the impact of media heterogeneity. Superdiffusion occurs when the position-dependent coefficient increases superlinearly, while subdiffusion occurs for sublinear and inverse power-law relations. This nontrivial interplay between heterogeneity and state-independent mechanisms also leads to anomalous yet Brownian, and non-Brownian yet Gaussian regimes. These findings emphasize the need for cautious interpretations of experiments and highlight the limitations of relying solely on mean squared displacements or position distributions for diffusion characterization.


Complexity of the COVID-19 pandemic in Maringá

Andre S. Sunahara, Arthur A. B. Pessa, Matjaž Perc, Haroldo V. Ribeiro
Scientific Reports 13, 12695 (2023)
Complex Systems

While extensive literature exists on the COVID-19 pandemic at regional and national levels, understanding its dynamics and consequences at the city level remains limited. This study investigates the pandemic in Maringá, a medium-sized city in Brazil’s South Region, using data obtained by actively monitoring the disease from March 2020 to June 2022. Despite prompt and robust interventions, COVID-19 cases increased exponentially during the early spread of COVID-19, with a reproduction number lower than that observed during the initial outbreak in Wuhan. Our research demonstrates the remarkable impact of non-pharmaceutical interventions on both mobility and pandemic indicators, particularly during the onset and the most severe phases of the emergency. However, our results suggest that the city’s measures were primarily reactive rather than proactive. Maringá faced six waves of cases, with the third and fourth waves being the deadliest, responsible for over two-thirds of all deaths and overwhelming the local healthcare system. Excess mortality during this period exceeded deaths attributed to COVID-19, indicating that the burdened healthcare system may have contributed to increased mortality from other causes. By the end of the fourth wave, nearly three-quarters of the city’s population had received two vaccine doses, significantly decreasing deaths despite the surge caused by the Omicron variant. Finally, we compare these findings with the national context and other similarly sized cities, highlighting substantial heterogeneities in the spread and impact of the disease.


Deep learning criminal networks

Haroldo V. Ribeiro, Diego D. Lopes, Arthur A.B. Pessa, Alvaro F. Martins, Bruno R. da Cunha, Sebastián Gonçalves, Ervin K. Lenzi, Quentin S. Hanley, Matjaž Perc
Chaos, Solitons & Fractals 172, 113579 (2023)
Complex NetworksCrimeMachine Learning

Recent advances in deep learning methods have enabled researchers to develop and apply algorithms for the analysis and modeling of complex networks. These advances have sparked a surge of interest at the interface between network science and machine learning. Despite this, the use of machine learning methods to investigate criminal networks remains surprisingly scarce. Here, we explore the potential of graph convolutional networks to learn patterns among networked criminals and to predict various properties of criminal networks. Using empirical data from political corruption, criminal police intelligence, and criminal financial networks, we develop a series of deep learning models based on the GraphSAGE framework that are able to recover missing criminal partnerships, distinguish among types of associations, predict the amount of money exchanged among criminal agents, and even anticipate partnerships and recidivism of criminals during the growth dynamics of corruption networks, all with impressive accuracy. Our deep learning models significantly outperform previous shallow learning approaches and produce high-quality embeddings for node and edge properties. Moreover, these models inherit all the advantages of the GraphSAGE framework, including the generalization to unseen nodes and scaling up to large graph structures.


Age and market capitalization drive large price variations of cryptocurrencies

Arthur A. B. Pessa, Matjaž Perc, Haroldo V. Ribeiro
Scientific Reports 13, 3351 (2023)

Cryptocurrencies are considered the latest innovation in finance with considerable impact across social, technological, and economic dimensions. This new class of financial assets has also motivated a myriad of scientific investigations focused on understanding their statistical properties, such as the distribution of price returns. However, research so far has only considered Bitcoin or at most a few cryptocurrencies, whilst ignoring that price returns might depend on cryptocurrency age or be influenced by market capitalization. Here, we therefore present a comprehensive investigation of large price variations for more than seven thousand digital currencies and explore whether price returns change with the coming-of-age and growth of the cryptocurrency market. We find that tail distributions of price returns follow power-law functions over the entire history of the considered cryptocurrency portfolio, with typical exponents implying the absence of characteristic scales for price variations in about half of them. Moreover, these tail distributions are asymmetric as positive returns more often display smaller exponents, indicating that large positive price variations are more likely than negative ones. Our results further reveal that changes in the tail exponents are very often simultaneously related to cryptocurrency age and market capitalization or only to age, with only a minority of cryptoassets being affected just by market capitalization or neither of the two quantities. Lastly, we find that the trends in power-law exponents usually point to mixed directions, and that large price variations are likely to become less frequent only in about 28% of the cryptocurrencies as they age and grow in market capitalization.



Effects of shady environments on fish collective behavior

Haroldo V. Ribeiro, Matthew R. Acre, Jacob D. Faulkner, Leonardo R. da Cunha, Katelyn M. Lawson, James J. Wamboldt, Marybeth K. Brey, Christa M. Woodley, Robin D. Calfee
Scientific Reports 12, 17873 (2022)
Collective BehaviorComplex NetworksFish

Despite significant efforts devoted to understanding the underlying complexity and emergence of collective movement in animal groups, the role of different external settings on this type of movement remains largely unexplored. Here, by combining time series analysis and complex network tools, we present an extensive investigation of the effects of shady environments on the behavior of a fish species (Silver Carp Hypophthalmichthys molitrix) within earthen ponds. We find that shade encourages fish residence during daylight hours, but the degree of preference for shade varies substantially among trials and ponds. Silver Carp are much slower and exhibit lower persistence in their speeds when under shade than out of it during daytime and nighttime, with fish displaying the highest persistence degree and speeds at night. Furthermore, our research shows that shade affects fish schooling behavior by reducing their polarization, number of interactions among individuals, and the stability among local neighbors; however, fish keep a higher local degree of order when under shade compared to nighttime positions.


Machine Learning Partners in Criminal Networks

D. D. Lopes, B. R. da Cunha, A. F. Martins, S. Goncalves, E. K. Lenzi, Q. S. Hanley, M. Perc, H. V. Ribeiro
Scientific Reports 12, 15746 (2022)
Complex NetworksCorruptionCrimeMachine Learning

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.


Schrödinger Equation with Geometric Constraints and Position-Dependent Mass

E. K. Lenzi, L. R. Evangelista, H. V. Ribeiro, R. L. Magin
Quantum Reports 4, 296 (2022).
Anomalous DiffusionFractional SchrödingerGeneralized Schrödinger

We investigate the solutions of a two-dimensional Schrödinger equation in the presence of geometric constraints, represented by a backbone structure with branches, by taking a position-dependent effective mass for each direction into account. We use Green’s function approach to obtain the solutions, which are given in terms of stretched exponential functions. The results can be linked to the properties of the system and show anomalous spreading for the wave packet. We also analyze the interplay between the backbone structure with branches constraining the different directions and the effective mass. In particular, we show how a fractional Schrödinger equation emerges from this scenario.


The Physics of Cities

Fabiano L. Ribeiro, Matjaž Perc, and Haroldo V. Ribeiro
Frontiers in Physics 10, 964701 (2022)
Complex SystemsEditorialScaling LawsUrban IndicatorsUrban Metrics

The word “physics” can be understood in at least two ways. First, based on the Greek origin of the word, physics means nature. Accordingly, when we state our intention to understand the physics of a phenomenon, we want to know how and why it behaves as it does. In other words, understanding the nature of something – be it a natural phenomenon or a concept – means that we comprehend the mechanisms governing the underlying system. The second meaning of the word “physics” is related to the field of study or knowledge area; that is, to all theoretical and experimental tools developed by physicists to understand our universe. These two meanings of the word “physics”, to some extent, appropriately describe a research area that has been highly active in recent years and is aimed at understanding the nature of cities through theoretical tools from physics. Many believe these studies mark the birth of a new discipline – a “new science of cities” – that aims to understand cities from the complexity science perspective.

Indeed, in the last few years, the physics community has made a breakthrough in understanding urban phenomena. In part, this ongoing progress is a consequence of a progressive and expressive increase in the amount of data available regarding the dynamics of urban life, but it is also driven by the necessity of unified theories to explain and propose experiments. These two elements, a massive volume of data and theories, which are so familiar to physicists, are essential for researchers to develop systematic ways to identify, describe, and explain commonalities, regularities, and universal patterns in cities – goals that were central to this Research Topic.

This special issue started with the article of Operti et al., which proposed a novel methodology to define and quantify the topography of racial residential segregation in large urban areas. By applying their method to New York City and using data from 1990 to 2010, Operti et al. investigated the dynamics of racial segregation for four racial categories as well as how it correlates with income, property values, and gentrification within neighborhoods. They reported that income inequality is higher in regions with high population densities of two or more races and that a positive flux of white citizens associates with an increase in property values, while the opposite is observed for places with a positive flux of black citizens. Furthermore, their approach also allowed the identification of the two largest displacements of black citizens and the emergence of gentrified regions.
The second contribution was made by Hayata, who focused on the empirical distributions of population, population density, and area of Japanese municipalities broken down into the 47 prefectures (regions or provinces) that constitute the country. Hayata used rank-size plots (also called Zipf plots) and tested different rank-size rules for the municipalities with a particular interest in urban areas. The author further considered that cities are competing for the extension of their areas and proposed a simple model inspired by sports tournaments, which in turn motivated an analysis of urban area evolution in the view of global warming and consequent shrinking of the land sizes.
Gere et al. contributed the third article of this collection where wealth distribution in villages of the commune Sâncraiu (Kalotaszentkirály) – a well-delimited region in Transylvania (Romania) – was studied using a unique dataset spanning radically different economic contexts: before collectivization imposed by communist (1961), the latest year of the communist regime in Romania (1989), and the current situation after over 30 years of the free-market economy (2021). Using an exactly solvable master equation, the authors were capable of realistically describing income dynamics and its distribution as well as discussing the observed socioeconomic changes. Among other findings, Gere et al. reported that the “rich gets richer” or the Matthew effect emerges with the fall of communism, which in turn is modeled as a linear growth rate.

In the fourth work of this research topic, Bassolas et al. also focused on spatial segregation, but instead of race, they investigated the spatial heterogeneity of different income categories in United States cities based on diffusion and synchronization dynamics occurring over a graph, where nodes are spatial units and connections indicate spatially adjacent units. Bassolas et al. used the time needed to reach the synchronization to quantify spatial heterogeneity, and among other findings, they reported that low- and high-income categories are more segregated in space than middle-income categories. Curiel, Cabrera-Arnau, and Bishop contributed the fifth article of this collection, in which the impact of city size on nearby cities was addressed using data from over two thousand African cities. The authors constructed the urban road network among these cities and proposed an approach to determine regions of influence of cities from this network. They then used urban scaling models to investigate the relationship between city size and characteristics of their regions of influence, finding that the size of a city impacts not only its urban indicators but also the indicators of neighboring cities. In particular, they reported that large cities drive urban emergence and growth of other cities even hundreds of kilometers apart.
Finally, the article by Molinero closed this special issue by presenting an analytical framework based on fractal theory to model urban growth. This theoretical approach was capable of describing many features of cities, including scaling laws with population size and the fractal nature of cities, and proved helpful in modeling urban growth with a case set on the United Kingdom system.

This Frontiers special issue has thus shed light on different urban phenomena through the unique lens of complexity science and physics. The published papers have contributed to the identification of novel regularities and connections between city properties, and we hope they will positively impact decision-making by public planners to optimize infrastructure and foster the economic development of urban areas.


Clustering free-falling paper motion with complexity and entropy

Arthur A. B. Pessa, Matjaz Perc, Haroldo V. Ribeiro
EPL 138, 30003 (2022)
Complex SystemsComplexity MeasureMachine LearningSymbolic Dynamics

Many simple natural phenomena are characterized by complex motion that appears random at first glance, but that often displays underlying patterns and behavior that can be clustered in groups. The movement of small pieces of paper falling through the air is one of these systems whose complete mathematical description seems unworkable. Understanding these types of motion thus demands automated experimentation capable of producing large datasets covering different behaviors — a task that has become feasible only recently with advances in computer vision and machine learning methods. Here we use one of these datasets related to the motion of free-falling paper with different shapes to propose an information-theoretical approach that automatically clusters different types of behavior. We evaluate the permutation entropy and statistical complexity from time series related to the observable area of free-falling paper pieces captured by a video camera. We find that chaotic and tumbling motions have a distinct average degree of entropy and complexity, allowing us to accurately discriminate between these two types of behavior with a simple unsupervised machine learning algorithm. Our method has a performance comparable to other approaches based on physical quantities but does not depend on reconstructing the three-dimensional falling trajectory.


Universality of political corruption networks

Alvaro F. Martins, Bruno R. da Cunha, Quentin S. Hanley, Sebastian Goncalves, Matjaz Perc, Haroldo V. Ribeiro
Scientific Reports 12, 6858 (2022)
Complex NetworksCrime

Corruption crimes demand highly coordinated actions among criminal agents to succeed. But research dedicated to corruption networks is still in its infancy and indeed little is known about the properties of these networks. Here we present a comprehensive investigation of corruption networks related to political scandals in Spain and Brazil over nearly three decades. We show that corruption networks of both countries share universal structural and dynamical properties, including similar degree distributions, clustering and assortativity coefficients, modular structure, and a growth process that is marked by the coalescence of network components due to a few recidivist criminals. We propose a simple model that not only reproduces these empirical properties but reveals also that corruption networks operate near a critical recidivism rate below which the network is entirely fragmented and above which it is overly connected. Our research thus indicates that actions focused on decreasing corruption recidivism may substantially mitigate this type of organized crime.


Permutation Jensen-Shannon distance: A versatile and fast symbolic tool for complex time-series analysis

Luciano Zunino, Felipe Olivares, Haroldo V. Ribeiro, Osvaldo A. Rosso
Physical Review E 105, 045310 (2022)
Complex SystemsComplexity MeasureData AnalysisSymbolic Dynamics

The main motivation of this paper is to introduce the permutation Jensen-Shannon distance, a symbolic tool able to quantify the degree of similarity between two arbitrary time series. This quantifier results from the fusion of two concepts, the Jensen-Shannon divergence and the encoding scheme based on the sequential ordering of the elements in the data series. The versatility and robustness of this ordinal symbolic distance for characterizing and discriminating different dynamics are illustrated through several numerical and experimental applications. Results obtained allow us to be optimistic about its usefulness in the field of complex time series analysis. Moreover, thanks to its simplicity, low computational cost, wide applicability and less susceptibility to outliers and artifacts, this ordinal measure can efficiently handle large amounts of data and help to tackle the current big data challenges.


Fractional Diffusion with Geometric Constraints: Application to Signal Decay in Magnetic Resonance Imaging (MRI)

Ervin K. Lenzi, Haroldo V. Ribeiro, Marcelo K. Lenzi, Luiz R. Evangelista, Richard L. Magin
Mathematics 10, 389 (2022)
Anomalous Diffusion

We investigate diffusion in three dimensions on a comb-like structure in which the particles move freely in a plane, but, out of this plane, are constrained to move only in the perpendicular direction. This model is an extension of the two-dimensional version of the comb model, which allows diffusion along the backbone when the particles are not in the branches. We also consider memory effects, which may be handled with different fractional derivative operators involving singular and non-singular kernels. We find exact solutions for the particle distributions in this model that display normal and anomalous diffusion regimes when the mean-squared displacement is determined. As an application, we use this model to fit the anisotropic diffusion of water along and across the axons in the optic nerve using magnetic resonance imaging. The results for the observed diffusion times (8 to 30 milliseconds) show an anomalous diffusion both along and across the fibers.


Population Density and Spreading of COVID-19 in England and Wales

Jack Sutton, Golnaz Shahtahmassebi, Haroldo V. Ribeiro, Quentin S. Hanley
PLoS ONE 7, e0261725 (2022)
Complex SystemsScaling LawsUrban IndicatorsUrban Metrics

We investigated daily COVID-19 cases and deaths in the 337 lower tier local authority regions in England and Wales to better understand how the disease propagated over a 15-month period. Population density scaling models revealed residual variance and skewness to be sensitive indicators of the dynamics of propagation. Lockdowns and schools reopening coincided with increased variance indicative of conditions with local impact and country scale heterogeneity. University reopening and December holidays reduced variance indicative of country scale homogenisation which reached a minimum in mid-January 2021. Homogeneous propagation was associated with better correspondence with normally distributed residuals while heterogeneous propagation was more consistent with skewed models. Skewness varied from strongly negative to strongly positive revealing an unappreciated feature of community propagation. Hot spots and super-spreading events are well understood descriptors of regional disease dynamics that would be expected to be associated with positively skewed distributions. Positively skewed behaviour was observed; however, negative skewness indicative of “cold-spots” and “super-isolation” dominated for approximately 8 months during the period of study. In contrast, death metrics showed near constant behaviour in scaling, variance, and skewness metrics over the full period with rural regions preferentially affected, an observation consistent with regional age demographics in England and Wales. Regional positions relative density scaling laws were remarkably persistent after the first 5-9 days of the available data set. The determinants of this persistent behaviour probably precede the pandemic and remain unchanged.


Determining liquid crystal properties with ordinal networks and machine learning

Arthur A. B. Pessa, Rafael S. Zola, Matjaz Perc, Haroldo V. Ribeiro
Chaos, Solitons & Fractals 154, 111607 (2022)
Complex SystemsComplexity MeasureLiquid CrystalMachine Learning

Machine learning methods are becoming increasingly important for the development of materials science. In spite of this, the use of image analysis in the development of these systems is still recent and underexplored, especially in materials often studied via optical imaging techniques such as liquid crystals. Here we apply the recently proposed method of ordinal networks to map optical textures obtained from experimental samples of liquid crystals into complex networks and use this representation jointly with a simple statistical learning algorithm to investigate different physical properties of these materials. Our research demonstrates that ordinal networks formed by only 24 nodes encode crucial information about liquid crystal properties, thus allowing us to train simple machine learning models capable of identifying and classifying mesophase transitions, distinguishing among different doping concentrations used to induce chiral mesophases, and predicting sample temperatures with outstanding accuracy. The precision and scalability of our approach indicate it can be used to probe properties of different materials in situations involving large-scale datasets or real-time monitoring systems.


Transient anomalous diffusion in heterogeneous media with stochastic resetting

M. K. Lenzi, E. K. Lenzi, L. M. S. Guilherme, L. R. Evangelista, H. V. Ribeiro
Physica A 588, 126560 (2022)
Anomalous DiffusionDiffusion

We investigate a diffusion process in heterogeneous media where particles stochastically reset to their initial positions at a constant rate. The heterogeneous media is modeled using a spatial-dependent diffusion coefficient with a power-law dependence on particles’ positions. We use the Green function approach to obtain exact solutions for the probability distribution of particles’ positions and the mean square displacement. These results are further compared and agree with numerical simulations of a Langevin equation. We also study the first-passage time problem associated with this diffusion process and obtain an exact expression for the mean first-passage time. Our findings show that this system exhibits non-Gaussian distributions, transient anomalous diffusion (sub- or superdiffusion) and stationary states that simultaneously depend on the media heterogeneity and the resetting rate. We further demonstrate that the media heterogeneity non-trivially affect the mean first-passage time, yielding an optimal resetting rate for which this quantity displays a minimum.



Commuting network effect on urban wealth scaling

Luiz G. A. Alves, Diego Rybski, Haroldo V. Ribeiro
Scientific Reports 11, 22918 (2021)
Complex NetworksComplex SystemsUrban IndicatorsUrban Metrics

Urban scaling theory explains the increasing returns to scale of urban wealth
indicators by the per capita increase of human interactions within cities. This
explanation implicitly assumes urban areas as isolated entities and ignores
their interactions. Here we investigate the effects of commuting networks on
the gross domestic product (GDP) of urban areas in the US and Brazil. We
describe the urban GDP as the output of a production process where population,
incoming commuters, and interactions between these quantities are the input
variables. This approach significantly refines the description of urban GDP and
shows that incoming commuters contribute to wealth creation in urban areas. Our
research indicates that changes in urban GDP related to proportionate changes
in population and incoming commuters depend on the initial values of these
quantities, such that increasing returns to scale are only possible when the
product between population and incoming commuters exceeds a well-defined


Sorption-Desorption, Surface diffusion, and Memory Effects in a 3D System

P. M. Ndiaye, F. W. Tavares, E. K. Lenzi, L. R. Evangelista, H. V. Ribeiro, D. Marin, L. M. S. Guilherme, R. S. Zola
J. Stat. Mech. (2021) 113202
Anomalous Diffusion

Bio and nature behaviors inspired modelling of diffusion and trapping of particles, key phenomena for life occurrence, must consider a myriad of ingredients, such as geometry, dimensionality, and scaled diffusion processes occurring across the bulk and surfaces. To attempt to approach this goal, we investigate the diffusion process in a system limited by a surface connected to the bulk through an especially designed boundary condition connected to some systems of interest, such as in living cells and bio-materials. The surface may sorb/desorb particles from the bulk and admits that the sorbed particles may diffuse within its structure, characterizing a lateral diffusion, before being desorbed back to the bulk. For this system, we find a wide variety of behavior by analyzing solutions obtained in terms of the Green function approach. The analytical calculation is checked against computer simulations, demonstrating a good agreement between analytical calculation and stochastic computer simulation.


Association between productivity and journal impact across disciplines and career age

Andre S. Sunahara, Matjaz Perc, Haroldo V. Ribeiro
Physical Review Research 3, 033158 (2021)
Complex SystemsScience of Science

The association between productivity and impact of scientific production is a long-standing debate in science that remains controversial and poorly understood. Here we present a large-scale analysis of the association between yearly publication numbers and average journal-impact metrics for the Brazilian scientific elite. We find this association to be discipline specific, career age dependent, and similar among researchers with outlier and nonoutlier performance. Outlier researchers either outperform in productivity or journal prestige, but they rarely do so in both categories. Nonoutliers also follow this trend and display negative correlations between productivity and journal prestige but with discipline-dependent intensity. Our research indicates that academics are averse to simultaneous changes in their productivity and journal-prestige levels over consecutive career years. We also find that career patterns concerning productivity and journal prestige are discipline-specific, having in common a raise of productivity with career age for most disciplines and a higher chance of outperforming in journal impact during early career stages.


ordpy: A Python package for data analysis with permutation entropy and ordinal network methods

Arthur A. B. Pessa and Haroldo V. Ribeiro
Chaos 31, 063110 (2021)
Complex NetworksComplex SystemsComplexity MeasureData AnalysisEntropySymbolic DynamicsTwo-dimensional Patterns

Since Bandt and Pompe’s seminal work, permutation entropy has been used in several applications and is now an essential tool for time series analysis. Beyond becoming a popular and successful technique, permutation entropy inspired a framework for mapping time series into symbolic sequences that triggered the development of many other tools, including an approach for creating networks from time series known as ordinal networks. Despite increasing popularity, the computational development of these methods is fragmented, and there were still no efforts focusing on creating a unified software package. Here, we present ordpy (, a simple and open-source Python module that implements permutation entropy and several of the principal methods related to Bandt and Pompe’s framework to analyze time series and two-dimensional data. In particular, ordpy implements permutation entropy, Tsallis and Rényi permutation entropies, complexity–entropy plane, complexity–entropy curves, missing ordinal patterns, ordinal networks, and missing ordinal transitions for one-dimensional (time series) and two-dimensional (images) data as well as their multiscale generalizations. We review some theoretical aspects of these tools and illustrate the use of ordpy by replicating several literature results.


Association between population distribution and urban GDP scaling

Haroldo V. Ribeiro, Milena Oehlers, Ana I. Moreno-Monroy, Jürgen P. Kropp, Diego Rybski
PLoS ONE 16, e0245771 (2021)
Scaling LawsUrban IndicatorsUrban Metrics

Urban scaling and Zipf’s law are two fundamental paradigms for the science of cities. These laws have mostly been investigated independently and are often perceived as disassociated matters. Here we present a large scale investigation about the connection between these two laws using population and GDP data from almost five thousand consistently-defined cities in 96 countries. We empirically demonstrate that both laws are tied to each other and derive an expression relating the urban scaling and Zipf exponents. This expression captures the average tendency of the empirical relation between both exponents, and simulations yield very similar results to the real data after accounting for random variations. We find that while the vast majority of countries exhibit increasing returns to scale of urban GDP, this effect is less pronounced in countries with fewer small cities and more metropolises (small Zipf exponent) than in countries with a more uneven number of small and large cities (large Zipf exponent). Our research puts forward the idea that urban scaling does not solely emerge from intra-city processes, as population distribution and scaling of urban GDP are correlated to each other.



Collective dynamics of stock market efficiency

Luiz G. A. Alves, Higor Y. D. Sigaki, Matjaz Perc, Haroldo V. Ribeiro
Scientific Reports 10, 21992 (2020)
Complex NetworksData Analysis

Summarized by the efficient market hypothesis, the idea that stock prices fully reflect all available information is always confronted with the behavior of real-world markets. While there is plenty of evidence indicating and quantifying the efficiency of stock markets, most studies assume this efficiency to be constant over time so that its dynamical and collective aspects remain poorly understood. Here we define the time-varying efficiency of stock markets by calculating the permutation entropy within sliding time-windows of log-returns of stock market indices. We show that major world stock markets can be hierarchically classified into several groups that display similar long-term efficiency profiles. However, we also show that efficiency ranks and clusters of markets with similar trends are only stable for a few months at a time. We thus propose a network representation of stock markets that aggregates their short-term efficiency patterns into a global and coherent picture. We find this financial network to be strongly entangled while also having a modular structure that consists of two distinct groups of stock markets. Our results suggest that stock market efficiency is a collective phenomenon that can drive its operation at a high level of informational efficiency, but also places the entire system under risk of failure.


Mapping images into ordinal networks

Arthur A. B. Pessa, Haroldo V. Ribeiro
Phys. Rev. E 102, 052312 (2020)
Complex NetworksData AnalysisTwo-dimensional Patterns

An increasing abstraction has marked some recent investigations in network science. Examples include the development of algorithms that map time series data into networks whose vertices and edges can have different interpretations, beyond the classical idea of parts and interactions of a complex system. These approaches have proven useful for dealing with the growing complexity and volume of diverse data sets. However, the use of such algorithms is mostly limited to one-dimension data, and there has been little effort towards extending these methods to higher-dimensional data such as images. Here we propose a generalization for the ordinal network algorithm for mapping images into networks. We investigate the emergence of connectivity constraints inherited from the symbolization process used for defining the network nodes and links, which in turn allows us to derive the exact structure of ordinal networks obtained from random images. We illustrate the use of this new algorithm in a series of applications involving randomization of periodic ornaments, images generated by two-dimensional fractional Brownian motion and the Ising model, and a data set of natural textures. These examples show that measures obtained from ordinal networks (such as average shortest path and global node entropy) extract important image properties related to roughness and symmetry, are robust against noise, and can achieve higher accuracy than traditional texture descriptors extracted from gray-level co-occurrence matrices in simple image classification tasks.


Rural-Urban Scaling of Age, Mortality, Crime and Property Reveals a Loss of Expected Self-Similar Behaviour

Jack Sutton, Golnaz Shahtahmassebi, Haroldo V. Ribeiro, Quentin S. Hanley
Scientific Reports 10, 16863 (2020)
Scaling LawsUrban IndicatorsUrban Metrics

The urban scaling hypothesis has improved our understanding of cities; however, rural areas have been neglected. We investigated rural-urban population density scaling in England and Wales using 67 indicators of crime, mortality, property, and age. Most indicators exhibited segmented scaling about a median critical density of 27 people per hectare. Above the critical density, urban regions preferentially attract young adults (25-40 yrs) and lose older people (>45 yrs). Density scale adjusted metrics (DSAMs) were analysed using hierarchical clustering, networks, and self-organizing maps (SOMs) revealing regional differences and an inverse relationship between excess value of property transactions and a range of preventable mortality (e.g. diabetes, suicide, lung cancer). The most striking finding is that age demographics break the expected self-similarity underlying the urban scaling hypothesis. Urban dynamism is fuelled by preferential attraction of young adults and not a fundamental property of total urban population.


City size and the spreading of COVID-19 in Brazil

Haroldo V. Ribeiro, Andre S. Sunahara, Jack Sutton, Matjaz Perc, Quentin S. Hanley
PLoS ONE 15, e0239699 (2020)
Scaling LawsUrban IndicatorsUrban Metrics

The current outbreak of the coronavirus disease 2019 (COVID-19) is an unprecedented example of how fast an infectious disease can spread around the globe (especially in urban areas) and the enormous impact it causes on public health and socio-economic activities. Despite the recent surge of investigations about different aspects of the COVID-19 pandemic, we still know little about the effects of city size on the propagation of this disease in urban areas. Here we investigate how the number of cases and deaths by COVID-19 scale with the population of Brazilian cities. Our results indicate small towns are proportionally more affected by COVID-19 during the initial spread of the disease, such that the cumulative numbers of cases and deaths per capita initially decrease with population size. However, during the long-term course of the pandemic, this urban advantage vanishes and large cities start to exhibit higher incidence of cases and deaths, such that every 1% rise in population is associated with a 0.14% increase in the number of fatalities per capita after about four months since the first two daily deaths. We argue that these patterns may be related to the existence of proportionally more health infrastructure in the largest cities and a lower proportion of older adults in large urban areas. We also find the initial growth rate of cases and deaths to be higher in large cities; however, these growth rates tend to decrease in large cities and to increase in small ones over time.


Anomalous diffusion and sorption-desorption process in complex fluid systems

F. W. Tavares, P. M. Ndiaye, E. K. Lenzi, L. R. Evangelista, H. V. Ribeiro, R. S. Zola
Commun. Nonlinear Sci. Numer. Simulat. 90 (2020) 105411
Anomalous Diffusion

Diffusion processes occurring in a myriad of systems sparkle great interest in understanding their general properties and applications. In this work, we investigate a broad set of diffusive systems that can be governed by a generalized diffusion equation and subjected to a surface that can promote sorption and, consequently, desorption, thus releasing the particles to the bulk. The general bulk equation used here can reproduce different diffusive regimes, among them, those described by the Cattaneo equation or by a fractional, anomalous diffusion. The equation related to the processes on the surface incorporates non-Debye relaxations which can be used to model non-exponential relaxations commonly found in biological or fractal systems. The solutions are obtained by using the Green function approach and show a rich class of behavior that can be related to anomalous diffusion.


Learning physical properties of liquid crystals with deep convolutional neural networks

Higor Y. D. Sigaki, Ervin K. Lenzi, Rafael S. Zola, Matjaz Perc, Haroldo V. Ribeiro
Scientific Reports 10, 7664 (2020)
Liquid CrystalMachine Learning

Machine learning algorithms have been available since the 1990s, but it is much more recently that they have come into use also in the physical sciences. While these algorithms have already proven to be useful in uncovering new properties of materials and in simplifying experimental protocols, their usage in liquid crystals research is still limited. This is surprising because optical imaging techniques are often applied in this line of research, and it is precisely with images that machine learning algorithms have achieved major breakthroughs in recent years. Here we use convolutional neural networks to probe several properties of liquid crystals directly from their optical images and without using manual feature engineering. By optimizing simple architectures, we find that convolutional neural networks can predict physical properties of liquid crystals with exceptional accuracy. We show that these deep neural networks identify liquid crystal phases and predict the order parameter of simulated nematic liquid crystals almost perfectly. We also show that convolutional neural networks identify the pitch length of simulated samples of cholesteric liquid crystals and the sample temperature of an experimental liquid crystal with very high precision.


Anomalous diffusion and random search in xyz-comb: Exact results

E. K. Lenzi, T. Sandev, H. V. Ribeiro, P. Jovanovski, A. Iomin and L. Kocarev
J. Stat. Mech. (2020) 053203
Anomalous DiffusionComb Model

We present an analytical treatment of anomalous diffusion in a three-dimensional comb (xyz-comb) by using the Green’s function approach. We derive exact analytical solutions for the propagators for an instantaneous point injection and natural boundary conditions. The marginal distributions for all three directions are obtained and the corresponding mean squared displacements are found. The analytical results are confirmed by numerical simulations in the framework of coupled Langevin equations. We also analyze a random search process on the xyz-comb, and analytical results on the first arrival time distribution, search reliability and efficiency are obtained. Results for multiple targets are presented as well. The developed approach can be useful for further studies of the Brownian–Lévy searches in the xyz-comb, and analysis of optimal search strategies on comb-like structures.


Quenched and Annealed Disorder Mechanisms in Comb-Models with Fractional Operators

A. A. Tateishi, H. V. Ribeiro, T. Sandev, I. Petreska, E. K. Lenz
Phys. Rev. E 101, 022135 (2020)
Anomalous DiffusionComb Model

Recent experimental findings on anomalous diffusion have demanded novel models that combine annealed (temporal) and quenched (spatial or static) disorder mechanisms. The comb model is a simplified description of diffusion on percolation clusters, where the comblike structure mimics quenched disorder mechanisms and yields a subdiffusive regime. Here we extend the comb model to simultaneously account for quenched and annealed disorder mechanisms. To do so, we replace usual derivatives in the comb diffusion equation by different fractional time-derivative operators and the conventional comblike structure by a generalized fractal structure. Our hybrid comb models thus represent a diffusion where different comblike structures describe different quenched disorder mechanisms, and the fractional operators account for various annealed disorder mechanisms. We find exact solutions for the diffusion propagator and mean square displacement in terms of different memory kernels used for defining the fractional operators. Among other findings, we show that these models describe crossovers from subdiffusion to Brownian or confined diffusions, situations emerging in empirical results. These results reveal the critical role of interactions between geometrical restrictions and memory effects on modeling anomalous diffusion.


Gender difference in candidature processes for Brazilian elections

M. Cardoso, R. S. Mendes, J. T. G. Souza, H. V. Ribeiro
Physica A 537, 122525 (2020)
Data AnalysisUrban IndicatorsUrban Metrics

Researchers of several areas have reported that there are still significant gender differences in their performances within different social systems, as in science and on-line communities, for example. This paper focuses on the gender difference in politic candidature processes, investigating the effect of the electorate size upon the candidate numbers, by considering the electorate of each Brazilian city and the respective number of candidates for Mayor and City Council Member (for women and men separately). We detected a sharp gap between the number of male candidates and number of female candidates, with the disadvantage occurring to greater intent in Mayoral elections. We also found non-linear mean correspondences, allometries. The allometric exponents display values very close to the City Council Member candidatures for women and men. However, they have notable smaller values for women than for men in Mayoral candidature. This shows a political hierarchy: the most influential position is related to the greater female underrepresentation.



Characterizing stochastic time series with ordinal networks

Arthur A. B. Pessa, Haroldo V. Ribeiro
Phys. Rev. E 100, 042304 (2019)
Complex NetworksComplex SystemsData Analysis

Approaches for mapping time series to networks have become essential tools for dealing with the increasing challenges of characterizing data from complex systems. Among the different algorithms, the recently proposed ordinal networks stand out due to their simplicity and computational efficiency. However, applications of ordinal networks have been mainly focused on time series arising from nonlinear dynamical systems, while basic properties of ordinal networks related to simple stochastic processes remain poorly understood. Here, we investigate several properties of ordinal networks emerging from random time series, noisy periodic signals, fractional Brownian motion, and earthquake magnitude series. For ordinal networks of random series, we present an approach for building the exact form of the adjacency matrix, which in turn is useful for detecting nonrandom behavior in time series and the existence of missing transitions among ordinal patterns. We find that the average value of a local entropy, estimated from transition probabilities among neighboring nodes of ordinal networks, is more robust against noise addition than the standard permutation entropy. We show that ordinal networks can be used for estimating the Hurst exponent of time series with accuracy comparable with state-of-the-art methods. Finally, we argue that ordinal networks can detect sudden changes in Earth’s seismic activity caused by large earthquakes.


Extensions and Solutions for Nonlinear Diffusion Equations and Random Walks

E. K. Lenzi, M. K. Lenzi, H. V. Ribeiro, L. R. Evangelista
Proc. Royal Soc. A 475, 20190432 (2019)
Anomalous Diffusion

We investigate a connection between random walks and nonlinear diffusion equations within the framework proposed by Einstein to explain the Brownian motion. We show here how to properly modify such framework in order to handle different physical scenarios. We obtain solutions for nonlinear diffusion equations by using the random walk approach and possible connections with a generalized thermostatistics formalism. Finally, we conclude that fractal and fractional derivatives may emerge in the context of nonlinear diffusion equations, depending on the choice of distribution functions related to the spreading of systems.


Effects of changing population or density on urban carbon dioxide emissions

Haroldo V. Ribeiro, Diego Rybski, Jürgen P. Kropp
Nature Communications 10, 3204 (2019).
Complex SystemsData AnalysisScaling LawsUrban IndicatorsUrban Metrics

The question of whether urbanization contributes to increasing carbon dioxide emissions has been mainly investigated via scaling relationships with population or population density. However, these approaches overlook the correlations between population and area, and ignore possible interactions between these quantities. Here, we propose a generalized framework that simultaneously considers the effects of population and area along with possible interactions between these urban metrics. Our results significantly improve the description of emissions and reveal the coupled role between population and density on emissions. These models show that variations in emissions associated with proportionate changes in population or density may not only depend on the magnitude of these changes but also on the initial values of these quantities. For US areas, the larger the city, the higher is the impact of changing its population or density on its emissions; but population changes always have a greater effect on emissions than population density.


Estimating physical properties from liquid crystal textures via machine learning and complexity-entropy methods

H. Y. D. Sigaki, R. F. de Souza, R. T. de Souza, R. S. Zola, H. V. Ribeiro
Phys. Rev. E 99, 013311 (2019).
Complexity MeasureData AnalysisLiquid CrystalMachine LearningTwo-dimensional Patterns

Imaging techniques are essential tools for inquiring a number of properties from different materials. Liquid crystals are often investigated via optical and image processing methods. In spite of that, considerably less attention has been paid to the problem of extracting physical properties of liquid crystals directly from textures images of these materials. Here we present an approach that combines two physics-inspired image quantifiers (permutation entropy and statistical complexity) with machine learning techniques for extracting physical properties of nematic and cholesteric liquid crystals directly from their textures images. We demonstrate the usefulness and accuracy of our approach in a series of applications involving simulated and experimental textures, in which physical properties of these materials (namely: average order parameter, sample temperature, and cholesteric pitch length) are predicted with significant precision. Finally, we believe our approach can be useful in more complex liquid crystal experiments as well as for probing physical properties of other materials that are investigated via imaging techniques.


Clustering patterns in efficiency and the coming-of-age of the cryptocurrency market

Higor Y. D. Sigaki, Matjaz Perc, Haroldo V. Ribeiro
Scientific Reports 9, 1440 (2019)
Complexity MeasureCryptocurrencyData AnalysisMachine Learning

The efficient market hypothesis has far-reaching implications for financial trading and market stability. Whether or not cryptocurrencies are informationally efficient has therefore been the subject of intense recent investigation. Here, we use permutation entropy and statistical complexity over sliding time-windows of price log returns to quantify the dynamic efficiency of more than four hundred cryptocurrencies. We consider that a cryptocurrency is efficient within a time-window when these two complexity measures are statistically indistinguishable from their values obtained on randomly shuffled data. We find that 37% of the cryptocurrencies in our study stay efficient over 80% of the time, whereas 20% are informationally efficient in less than 20% of the time. Our results also show that the efficiency is not correlated with the market capitalization of the cryptocurrencies. A dynamic analysis of informational efficiency over time reveals clustering patterns in which different cryptocurrencies with similar temporal patterns form four clusters, and moreover, younger currencies in each group appear poised to follow the trend of their ‘elders’. The cryptocurrency market thus already shows notable adherence to the efficient market hypothesis, although data also reveals that the coming-of-age of digital currencies is in this regard still very much underway.


The hidden traits of endemic illiteracy in cities

Luiz G. A. Alves, Jose S. Andrade Jr., Quentin S. Hanley, Haroldo V. Ribeiro
Physica A 515, 566-574 (2019)
Complex SystemsUrban IndicatorsUrban Metrics

In spite of the considerable progress towards reducing illiteracy rates, many countries, including developed ones, have encountered difficulty achieving further reduction in these rates. This is worrying because illiteracy has been related to numerous health, social, and economic problems. Here, we show that the spatial patterns of illiteracy in urban systems have several features analogous to the spread of diseases such as dengue and obesity. Our results reveal that illiteracy rates are spatially long-range correlated, displaying non-trivial clustering structures characterized by percolation-like transitions and fractality. These patterns can be described in the context of percolation theory of long-range correlated systems at criticality. Together, these results provide evidence that the illiteracy incidence can be related to a transmissible process, in which the lack of access to minimal education propagates in a population in a similar fashion to endemic diseases.



History of art paintings through the lens of entropy and complexity

Higor Y. D. Sigaki, Matjaz Perc, Haroldo V. Ribeiro
Proc. Natl. Acad. Sci. U.S.A Proc. Natl. Acad. Sci. U.S.A 155, E8585-E8594 (2018).
ArtComplex SystemsData Analysis

Art is the ultimate expression of human creativity that is deeply influenced by the philosophy and culture of the corresponding historical epoch. The quantitative analysis of art is therefore essential for better understanding human cultural evolution. Here, we present a large-scale quantitative analysis of almost 140,000 paintings, spanning nearly a millennium of art history. Based on the local spatial patterns in the images of these paintings, we estimate the permutation entropy and the statistical complexity of each painting. These measures map the degree of visual order of artworks into a scale of order–disorder and simplicity–complexity that locally reflects qualitative categories proposed by art historians. The dynamical behavior of these measures reveals a clear temporal evolution of art, marked by transitions that agree with the main historical periods of art. Our research shows that different artistic styles have a distinct average degree of entropy and complexity, thus allowing a hierarchical organization and clustering of styles according to these metrics. We have further verified that the identified groups correspond well with the textual content used to qualitatively describe the styles and the applied complexity–entropy measures can be used for an effective classification of artworks.


A nonlinear Fokker-Planck equation approach for interacting systems: Anomalous diffusion and Tsallis statistics

D. Marin, M.A. Ribeiro, H.V. Ribeiro, E.K. Lenzi
Physics Letters A 382, 1903 (2018)
Anomalous Diffusion

We investigate the solutions for a set of coupled nonlinear Fokker–Planck equations coupled by the diffusion coefficient in presence of external forces. The coupling by the diffusion coefficient implies that the diffusion of each species is influenced by the other and vice versa due to this term, which represents an interaction among them. The solutions for the stationary case are given in terms of the Tsallis distributions, when arbitrary external forces are considered. We also use the Tsallis distributions to obtain a time dependent solution for a linear external force. The results obtained from this analysis show a rich class of behavior related to anomalous diffusion, which can be characterized by compact or long-tailed distributions.


Crime prediction through urban metrics and statistical learning

Luiz G. A. Alves, Haroldo V. Ribeiro, Francisco A. Rodrigues
Physica A 505, 435 (2018)
CrimeMachine Learning

Understanding the causes of crime is a longstanding issue in researcher’s agenda. While it is a hard task to extract causality from data, several linear models have been proposed to predict crime through the existing correlations between crime and urban metrics. However, because of non-Gaussian distributions and multicollinearity in urban indicators, it is common to find controversial conclusions about the influence of some urban indicators on crime. Machine learning ensemble-based algorithms can handle well such problems. Here, we use a random forest regressor to predict crime and quantify the influence of urban indicators on homicides. Our approach can have up to 97\% of accuracy on crime prediction, and the importance of urban indicators is ranked and clustered in groups of equal influence, which are robust under slightly changes in the data sample analyzed. Our results determine the rank of importance of urban indicators to predict crime, unveiling that unemployment and illiteracy are the most important variables for describing homicides in Brazilian cities. We further believe that our approach helps in producing more robust conclusions regarding the effects of urban indicators on crime, having potential applications for guiding public policies for crime control.


Nonlinear Diffusion Equation with Reaction Terms: Analytical and Numerical Results

E. K. Lenzi, M. A. Ribeiro, M. E. K. Fuziki, M. K. Lenzi, H. V. Ribeiro
Applied Mathematics and Computation 330, 254-265 (2018)
Anomalous Diffusion

We investigate a process obtained from a combination of nonlinear diffusion equations with reaction terms connected to a reversible process, i.e., 1 -> 2, of two species. This feature implies that the species 1 reacts producing the species 2, and vice-versa. A particular case emerging from this scenario is represented by 1 -> 2 (or 2 ->1), characterizing an irreversible process where one species produces the other. The results show that in the asymptotic limit of small and long times the behavior of the species is essentially governed by the diffusive term terms. For intermediate times, the behavior of the system depends on the reaction terms, particularly on the rates related to the reaction terms. In the presence of external forces, significant changes occur in the asymptotic limits. For these cases, we relate the solutions with the q-exponential function of the Tsallis statistic to highlight the compact or long-tailed behavior of the solutions and to establish a connection with the Tsallis thermo-statistic. We also extend the results to the spatial fractional differential operator by considering long-tailed distributions for the probability density function.


Unveiling Relationships Between Crime and Property in England and Wales Via Density Scale-Adjusted Metrics and Network Tools

Haroldo V. Ribeiro, Quentin S. Hanley, Dan Lewis
PLoS ONE 13, e0192931 (2018)
Complex SystemsCrimeUrban IndicatorsUrban Metrics

Scale-adjusted metrics (SAMs) are a significant achievement of the urban scaling hypothesis. SAMs remove the inherent biases of per capita measures computed in the absence of isometric allometries. However, this approach is limited to urban areas, while a large portion of the world’s population still lives outside cities and rural areas dominate land use worldwide. Here, we extend the concept of SAMs to population density scale-adjusted metrics (DSAMs) to reveal relationships among different types of crime and property metrics. Our approach allows all human environments to be considered, avoids problems in the definition of urban areas, and accounts for the heterogeneity of population distributions within urban regions. By combining DSAMs, cross-correlation, and complex network analysis, we find that crime and property types have intricate and hierarchically organized relationships leading to some striking conclusions. Drugs and burglary had uncorrelated DSAMs and, to the extent property transaction values are indicators of affluence, twelve out of fourteen crime metrics showed no evidence of specifically targeting affluence. Burglary and robbery were the most connected in our network analysis and the modular structures suggest an alternative to “zero-tolerance” policies by unveiling the crime and/or property types most likely to affect each other.


Characterization of Time Series Via Rényi Complexity-Entropy Curves

Max Jauregui, Luciano Zunino, Ervin K. Lenzi, Renio S. Mendes, Haroldo V. Ribeiro
Physica A 498, 74 (2018)
Complex SystemsComplexity MeasureData AnalysisSymbolic Dynamics

One of the most useful tools for distinguishing between chaotic and stochastic time series is the so-called complexity-entropy causality plane. This diagram involves two complexity measures: the Shannon entropy and the statistical complexity. Recently, this idea has been generalized by considering the Tsallis monoparametric generalization of the Shannon entropy, yielding complexity-entropy curves. These curves have proven to enhance the discrimination among different time series related to stochastic and chaotic processes of numerical and experimental nature. Here we further explore these complexity-entropy curves in the context of the Rényi entropy, which is another monoparametric generalization of the Shannon entropy. By combining the Rényi entropy with the proper generalization of the statistical complexity, we associate a parametric curve (the Rényi complexity-entropy curve) with a given time series. We explore this approach in a series of numerical and experimental applications, demonstrating the usefulness of this new technique for time series analysis. We show that the R\’enyi complexity-entropy curves enable the differentiation among time series of chaotic, stochastic, and periodic nature. In particular, time series of stochastic nature are associated with curves displaying positive curvature in a neighborhood of their initial points, whereas curves related to chaotic phenomena have a negative curvature; finally, periodic time series are represented by vertical straight lines.


The dynamical structure of political corruption networks

Haroldo V. Ribeiro, Luiz G. A. Alves, Alvaro F. Martins, Ervin K. Lenzi, Matjaz Perc
Journal of Complex Networks, cny002 (2018)
Complex NetworksCorruptionData Analysis

Corruptive behaviour in politics limits economic growth, embezzles public funds, and promotes socio-economic inequality in modern democracies. We analyse well-documented political corruption scandals over the past 27 years, focusing on the dynamical structure of networks where two individuals are connected if they were involved in the same scandal. Our research reveals that corruption runs in small groups that rarely comprise more than eight people, in networks that have hubs and a modular structure that encompasses more than one corruption scandal. We observe abrupt changes in the size of the largest connected component and in the degree distribution, which are due to the coalescence of different modules when new scandals come to light or when governments change. We show further that the dynamical structure of political corruption networks can be used for successfully predicting partners in future scandals. We discuss the important role of network science in detecting and mitigating political corruption.



The role of fractional time-derivative operators on anomalous diffusion

A. A. Tateishi, H. V. Ribeiro, E. K. Lenzi
Front. Phys. 5, 52 (2017)
Anomalous DiffusionDiffusion

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


Fractional Calculus in Electrical Impedance Spectroscopy: Poisson-Nernst-Planck model and Extensions

E. K. Lenzi, R. S. Zola, H. V. Ribeiro, L. R. Evangelista
Int. J. Electrochem. Sci., 12 (2017) 11677-11691
Electrical Response

We review some analytical results obtained in the context of the fractional calculus for the electrical spectroscopy impedance, a technique usually employed to interpret experimental data regarding the electrical response of an electrolytic cell. We start by reviewing the main points of the standard Poisson – Nernst – Planck model. After, we present an extension that incorporates fractional time derivatives of distributed order to the diffusion equation. Then, we include fractional time derivatives on the boundary conditions in order to face the problems that are characterized, in the low-frequency limit, by a frequency dispersion and, consequently, leads to a response in the form , where is the electrical impedance, , with being the frequency of the applied voltage, and . This scenario is extended in order to encompass also the systems characterized by Ohmic electrodes. For these cases, by focusing the low-frequency regime, we discuss the applicability of such extensions as a tool to describe experimental data. This analysis is applied in the description of the electrical impedance of electrolytic cells with Milli – Q water and a weak electrolytic solution of KCl.


Characterizing Time Series via Complexity-Entropy Curves

Haroldo V. Ribeiro, Max Jauregui, Luciano Zunino, Ervin K. Lenzi
Physical Review E 95, 062106 (2017)
Complex SystemsComplexity MeasureData AnalysisSymbolic Dynamics

The search for patterns in time series is a very common task when dealing with complex systems. This is usually accomplished by employing a complexity measure such as entropies and fractal dimensions. However, such measures usually only capture a single aspect of the system dynamics. Here, we propose a family of complexity measures for time series based on a generalization of the complexity-entropy causality plane. By replacing the Shannon entropy by a monoparametric entropy (Tsallis q-entropy) and after considering the proper generalization of the statistical complexity (q-complexity), we build up a parametric curve (the q-complexity-entropy curve) that is used for characterizing and classifying time series. Based on simple exact results and numerical simulations of stochastic processes, we show that these curves can distinguish among different long-range, short-range, and oscillating correlated behaviors. Also, we verify that simulated chaotic and stochastic time series can be distinguished based on whether these curves are open or closed. We further test this technique in experimental scenarios related to chaotic laser intensity, stock price, sunspot, and geomagnetic dynamics, confirming its usefulness. Finally, we prove that these curves enhance the automatic classification of time series with long-range correlations and interbeat intervals of healthy subjects and patients with heart disease.


Ion Motion in Electrolytic Cells: Anomalous Diffusion Evidences

E. K. Lenzi, R. S. Zola, H. V. Ribeiro, Denner S. Vieira, F. Ciuchi∥, A. Mazzulla , N. Scaramuzza, and L. R. Evangelista
J. Phys. Chem. B 12 (2017) 2882–2886
Anomalous DiffusionElectrical Response

In this study, we argue that ion motion in electrolytic cells containing Milli-Q water, weak electrolytes, or liquid crystals may exhibit unusual diffusive regimes that deviate from the expected behavior, leading the system to present an anomalous diffusion. Our arguments lie on the investigation of the electrical conductivity and its relationship with the mean square displacement, which may be used to characterize the ionic motion. In our analysis, the Poisson–Nernst–Planck diffusional model is used with extended boundary conditions to simulate the charge transfer, accumulation, and/or adsorption–desorption at the electrode surfaces.


Intermittent Motion, Nonlinear Diffusion Equation and Tsallis Formalism

E. K. Lenzi, L. R. da Silva, M. K. Lenzi, M. A. F. dos Santos, H. V. Ribeiro, L. R. Evangelista
Entropy 19, 42 (2017)
Generalized Schrödinger

We investigate an intermittent process obtained from the combination of a nonlinear
diffusion equation and pauses. We consider the porous media equation with reaction terms related to
the rate of switching the particles from the diffusive mode to the resting mode or switching them from
the resting to the movement. The results show that in the asymptotic limit of small and long times,
the spreading of the system is essentially governed by the diffusive term. The behavior exhibited for
intermediate times depends on the rates present in the reaction terms. In this scenario, we show that,
in the asymptotic limits, the distributions for this process are given by in terms of power laws which
may be related to the q-exponential present in the Tsallis statistics. Furthermore, we also analyze a
situation characterized by different diffusive regimes, which emerges when the diffusive term is a
mixing of linear and nonlinear terms.


Asymptotic behaviors of the Poisson-Nernst-Planck model, generalizations and best adjust of experimental data

E.K. Lenzi, R.S. Zola, R. Rossato, H.V. Ribeiro, D.S. Vieira, L.R. Evangelista
Electrochimica Acta 226, 40 (2017)
Anomalous DiffusionElectrical Response

We analyze the asymptotic behavior of the impedance (or immittance) spectroscopy response of an electrolytic cell in a finite-length situation obtained from the Poisson-Nernst-Planck (PNP) diffusional model and extensions by taking into account different surface effects. The analysis starts with the case characterized by perfect blocking electrodes and proceeds by considering non-blocking conditions on electrodes surface. We argue that the imaginary part of the impedance may be directly related to the boundary condition on the electrode surface, such as charge accumulation and/or transfer by electrochemical reaction or adsorption-desorption processes. We also compare the theoretical predictions with experimental data obtained for a weak electrolytic solution of KClO3.



Anomalous diffusion and transport in heterogeneous systems separated by a membrane

E. K. Lenzi, H. V. Ribeiro, A. A. Tateishi, R. S. Zola, L. R. Evangelista
Proc. R. Soc. A 472, 20160502 (2016)
Anomalous Diffusion

Diffusion of particles in a heterogeneous system separated by a semipermeable membrane is investigated. The particle dynamics is governed by fractional diffusion equations in the bulk and by kinetic equations on the membrane, which characterizes an interface between two different media. The kinetic equations are solved by incorporating memory effects to account for anomalous diffusion and, consequently, non-Debye relaxations. A rich variety of behaviours for the particle distribution at the interface and in the bulk may be found, depending on the choice of characteristic times in the boundary conditions and on the fractional index of the modelling equations.


Differences in Collaboration Patterns across Discipline, Career Stage, and Gender

Xiao Han T. Zeng, Jordi Duch, Marta Sales-Pardo, João A. G. Moreira, Filippo Radicchi, Haroldo V. Ribeiro, Teresa K. Woodruff, Luís A. Nunes Amaral
PLOS Biology 14, e1002573
GenderScience of Science

Collaboration plays an increasingly important role in promoting research productivity and impact. What remains unclear is whether female and male researchers in science, technology, engineering, and mathematical (STEM) disciplines differ in their collaboration propensity. Here, we report on an empirical analysis of the complete publication records of 3,980 faculty members in six STEM disciplines at select U.S. research universities. We find that female faculty have significantly fewer distinct co-authors over their careers than males, but that this difference can be fully accounted for by females’ lower publication rate and shorter career lengths. Next, we find that female scientists have a lower probability of repeating previous co-authors than males, an intriguing result because prior research shows that teams involving new collaborations produce work with higher impact. Finally, we find evidence for gender segregation in some sub-disciplines in molecular biology, in particular in genomics where we find female faculty to be clearly under-represented.


Discriminating image textures with the multiscale two-dimensional complexity-entropy causality plane

Luciano Zunino, Haroldo V. Ribeiro
Chaos, Solitons & Fractals 91, 679 (2016)
Complexity MeasureTwo-dimensional Patterns

The aim of this paper is to further explore the usefulness of the two-dimensional complexity-entropy causality plane as a texture image descriptor. A multiscale generalization is introduced in order to distinguish between different roughness features of images at small and large spatial scales. Numerically generated two-dimensional structures are initially considered for illustrating basic concepts in a controlled framework. Then, more realistic situations are studied. Obtained results allow us to confirm that intrinsic spatial correlations of images are successfully unveiled by implementing this multiscale symbolic information-theory approach. Consequently, we conclude that the proposed representation space is a versatile and practical tool for identifying, characterizing and discriminating image textures.


Extensive Characterization of Seismic Laws in Acoustic Emissions of Crumpled Plastic Sheets

Leandro S. Costa, Ervin K. Lenzi, Renio S. Mendes, Haroldo V. Ribeiro
EPL, 114 (2016) 59002
Cracking NoiseData Analysis

Statistical similarities between earthquakes and other systems that emit cracking noises have been explored in diverse contexts, ranging from materials science to financial and social systems. Such analogies give promise of a unified and universal theory for describing the complex responses of those systems. There are, however, very few attempts to simultaneously characterize the most fundamental seismic laws in such systems. Here we present a complete description of the Gutenberg-Richter law, the recurrence times, Omori’s law, the productivity law, and Bath’s law for the acoustic emissions that happen in the relaxation process of uncrumpling thin plastic sheets. Our results show that these laws also appear in this phenomenon, but (for most cases) with different parameters from those reported for earthquakes and fracture experiments. This study thus contributes to elucidate the parallel between seismic laws and cracking noises in uncrumpling processes, revealing striking qualitative similarities but also showing that these processes display unique features.


The Advantage of Playing Home in NBA: Microscopic, Team-Specific and Evolving Features

Haroldo V. Ribeiro, Satyam Mukherjee , Xiao Han T. Zeng
PLoS ONE 11, e0152440 (2016)
Complex SystemsNBA

The idea that the success rate of a team increases when playing home is broadly accepted and documented for a wide variety of sports. Investigations on the so-called “home advantage phenomenon” date back to the 70’s and ever since has attracted the attention of scholars and sport enthusiasts. These studies have been mainly focused on identifying the phenomenon and trying to correlate it with external factors such as crowd noise and referee bias. Much less is known about the effects of home advantage in the “microscopic” dynamics of the game (within the game) or possible team-specific and evolving features of this phenomenon. Here we present a detailed study of these previous features in the National Basketball Association (NBA). By analyzing play-by-play events of more than sixteen thousand games that span thirteen NBA seasons, we have found that home advantage affects the microscopic dynamics of the game by increasing the scoring rates and decreasing the time intervals between scores of teams playing home. We verified that these two features are different among the NBA teams, for instance, the scoring rate of the Cleveland Cavaliers team is increased ≈0.16 points per minute (on average the seasons 2004–05 to 2013–14) when playing home, whereas for the New Jersey Nets (now the Brooklyn Nets) this rate increases in only ≈0.04 points per minute. We further observed that these microscopic features have evolved over time in a non-trivial manner when analyzing the results team-by-team. However, after averaging over all teams some regularities emerge; in particular, we noticed that the average differences in the scoring rates and in the characteristic times (related to the time intervals between scores) have slightly decreased over time, suggesting a weakening of the phenomenon. This study thus adds evidence of the home advantage phenomenon and contributes to a deeper understanding of this effect over the course of games.


Transient superdiffusion and long-range correlations in the motility patterns of trypanosomatid flagellate protozoa

L. G. A. Alves, D. B. Scariot, R. R. Guimarães, C. V. Nakamura, R. S. Mendes, H. V. Ribeiro
PLoS ONE 11, e0152092 (2016)
Anomalous DiffusionDiffusion

We report on a diffusive analysis of the motion of flagellate protozoa species. These parasites are the etiological agents of neglected tropical diseases: leishmaniasis caused by Leishmania amazonensis and Leishmania braziliensis, African sleeping sickness caused by Trypanosoma brucei, and Chagas disease caused by Trypanosoma cruzi. By tracking the positions of these parasites and evaluating the variance related to the radial positions, we find that their motions are characterized by a short-time transient superdiffusive behavior. Also, the probability distributions of the radial positions are self-similar and can be approximated by a stretched Gaussian distribution. We further investigate the probability distributions of the radial velocities of individual trajectories. Among several candidates, we find that the generalized gamma distribution shows a good agreement with these distributions. The velocity time series have long-range correlations, displaying a strong persistent behavior (Hurst exponents close to one). The prevalence of “universal” patterns across all analyzed species indicates that similar mechanisms may be ruling the motion of these parasites, despite their differences in morphological traits. In addition, further analysis of these patterns could become a useful tool for investigating the activity of new candidate drugs against these and others neglected tropical diseases.


Rural to urban population density scaling of crime and property transactions in English and Welsh Parliamentary Constituencies

Quentin S. Hanley, Dan Lewis, Haroldo V. Ribeiro
PLoS ONE 11, e0149546 (2016)
CrimeScaling LawsUrban IndicatorsUrban Metrics

Urban population scaling of resource use, creativity metrics, and human behaviors has been widely studied. These studies have not looked in detail at the full range of human environments which represent a continuum from the most rural to heavily urban. We examined monthly police crime reports and property transaction values across all 573 Parliamentary Constituencies in England and Wales, finding that scaling models based on population density provided a far superior framework to traditional population scaling. We found four types of scaling: i) non-urban scaling in which a single power law explained the relationship between the metrics and population density from the most rural to heavily urban environments, ii) accelerated scaling in which high population density was associated with an increase in the power-law exponent, iii) inhibited scaling where the urban environment resulted in a reduction in the power-law exponent but remained positive, and iv) collapsed scaling where transition to the high density environment resulted in a negative scaling exponent. Urban scaling transitions, when observed, took place universally between 10 and 70 people per hectare. This study significantly refines our understanding of urban scaling, making clear that some of what has been previously ascribed to urban environments may simply be the high density portion of non-urban scaling. It also makes clear that some metrics undergo specific transitions in urban environments and these transitions can include negative scaling exponents indicative of collapse. This study gives promise of far more sophisticated scale adjusted metrics and indicates that studies of urban scaling represent a high density subsection of overall scaling relationships which continue into rural environments.


Fractional diffusion equations coupled by reaction terms

E.K. Lenzi, R. Menechini Neto, A.A. Tateishi, M.K. Lenzi, H.V. Ribeiro
Physica A 458, 9 (2016)
Anomalous Diffusion

We investigate the behavior for a set of fractional reaction–diffusion equations that extend the usual ones by the presence of spatial fractional derivatives of distributed order in the diffusive term. These equations are coupled via the reaction terms which may represent reversible or irreversible processes. For these equations, we find exact solutions and show that the spreading of the distributions is asymptotically governed by the same the long-tailed distribution. Furthermore, we observe that the coupling introduced by reaction terms creates an interplay between different diffusive regimes leading us to a rich class of behaviors related to anomalous diffusion.


Characterization of river flow fluctuations via horizontal visibility graphs

A. C. Braga, L. G. A. Alves, L. S. Costa, A. A. Ribeiro, M. M. A. de Jesus, A. A. Tateishi, H. V. Ribeiro
Physica A 444, 1003 (2016)
Complex NetworksComplex SystemsData Analysis

We report on a large-scale characterization of river discharges by employing the network framework of the horizontal visibility graph. By mapping daily time series from 141 different stations of 53 Brazilian rivers into complex networks, we present a useful approach for investigating the dynamics of river flows. We verified that the degree distributions of these networks were well described by exponential functions, where the characteristic exponents are almost always larger than the value obtained for random time series. The faster-than-random decay of the degree distributions is an another evidence that the fluctuation dynamics underlying the river discharges has a long-range correlated nature. We further investigated the evolution of the river discharges by tracking the values of the characteristic exponents (of the degree distribution) and the global clustering coefficients of the networks over the years. We show that the river discharges in several stations have evolved to become more or less correlated (and displaying more or less complex internal network structures) over the years, a behavior that could be related to changes in the climate system and other man-made phenomena.


Solutions for a sorption process governed by a fractional diffusion equation

E.K. Lenzi, M.A.F. dos Santos, D.S. Vieira, R.S. Zolac, H.V. Ribeiro
Physica A 443, 32 (2016)
Anomalous DiffusionDiffusion

We investigate a sorption process where one substance spreads out through another having possibility of chemical reaction between them. So as to describe this process, we have considered the bulk dynamics governed by a fractional diffusion equation, where the reaction term may describe an irreversible or a reversible process. This reaction term represents a generalization of the first order kinetic equation taking memory effects into account. The analytical solutions for the mean square displacement, survival probability and probability density of the particles we have obtained show a rich class of behaviors connected to anomalous diffusion.



Scale-adjusted metrics for predicting the evolution of urban indicators and quantifying the performance of cities

L. G. A. Alves, E. K. Lenzi, R. S. Mendes, and H. V. Ribeiro
PLoS ONE 10, e0134862 (2015)
Complex SystemsCrimeUrban IndicatorsUrban Metrics

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.


Spatial correlations, clustering and percolation-like transitions in homicide crimes

L.G.A. Alves, E.K. Lenzi, R.S. Mendes, and H.V. Ribeiro
EPL 111, 18002 (2015)
Complex SystemsCrimeUrban IndicatorsUrban Metrics

The spatial dynamics of criminal activities has been recently studied through statistical physics methods; however, models and results have been focused on local scales (city level) and much less is known about these patterns at larger scales such as at a country level. Here we report on a characterization of the spatial dynamics of the homicide crimes along the Brazilian territory using data from all cities (~5000) in a period of more than thirty years. Our results show that the spatial correlation function in the per capita homicides decays exponentially with the distance between cities and that the characteristic correlation length displays an acute increasing trend in the latest years. We also investigate the formation of spatial clusters of cities via a percolation-like analysis, where clustering of cities and a phase transition-like behavior describing the size of the largest cluster as a function of a homicide threshold are observed. This transition-like behavior presents evolutive features characterized by an increasing in the homicide threshold (where the transitions occur) and by a decreasing in the transition magnitudes (length of the jumps in the cluster size). We believe that our work sheds new lights on the spatial patterns of criminal activities at large scales, which may contribute for better political decisions and resources allocation as well as opens new possibilities for modeling criminal activities by setting up fundamental empirical patterns at large scales.


Analogies between the cracking noise of ethanol-dampened charcoal and earthquakes

H. V. Ribeiro, L. S. Costa, L. G. A. Alves, P. A. Santoro, S. Picoli, E. K. Lenzi, R. S. Mendes
Phys. Rev. Lett. 115, 025503 (2015)
Cracking NoiseEarthquakes

We report on an extensive characterization of the cracking noise produced by charcoal samples when dampened with ethanol. We argue that the evaporation of ethanol causes transient and irregularly distributed internal stresses that promote the fragmentation of the samples and mimic some situations found in mining processes. The results show that, in general, the most fundamental seismic laws ruling earthquakes (the Gutenberg-Richter law, the unified scaling law for the recurrence times, Omori’s law, the productivity law, and Båth’s law) hold under the conditions of the experiment. Some discrepancies were also identified (a smaller exponent in the Gutenberg-Richter law, a stationary behavior in the aftershock rates for long times, and a double power-law relationship in the productivity law) and are related to the different loading conditions. Our results thus corroborate and elucidate the parallel between the seismic laws and fracture experiments caused by a more complex loading condition that also occurs in natural and induced seismicity (such as long-term fluid injection and gas-rock outbursts in mining processes).


Electrolytic cell containing different groups of ions with anomalous diffusion approach

F.R.G.B. Silva, R. Rossato, E.K. Lenzi, R.S. Zola, H.V. Ribeiro, M.K. Lenzi, G. Gonçalves
Journal of Electroanalytical Chemistry 746 (2015) 25–30
Anomalous DiffusionElectrical Response

The electrical response of an electrolytic cell containing more than one group of ions is investigated under the fractional approach where integro – differential boundary conditions and fractional time derivative of distributed order are considered. The model derived here, which accounts for anomalous diffusion of charges in a dielectric media, is compared with experimental data for mixtures of two salts with same valence and water and a good agreement was found. We show that in the low frequency limit, the electrical response is essentially governed by the boundary conditions and suppress the formation of a second plateau predicted when surface effects are neglected, being therefore linked to an anomalous diffusive process which, in the usual circuit description, may be connected to constant phase elements. Our model may be important for the interpretation of ionic density measurements in liquid crystals and other electrolytic cells in a more realistic fashion.


Solutions for a q-generalized Schrödinger equation of entangled interacting particles

Luiz G. A. Alves, Haroldo V. Ribeiro, Maike A. F. Santos, Renio S. Mendes, Ervin K. Lenzi
Physica A 429, 35 (2015)
Generalized Schrödinger

We report on the time dependent solutions of the q-generalized Schrödinger equation proposed by Nobre et al. (2011). Here we investigate the case of two free particles and also the case where two particles were subjected to a Moshinsky-like potential with time dependent coefficients. We work out analytical and numerical solutions for different values of the parameter q and also show that the usual Schrödinger equation is recovered in the limit of q → 1. An intriguing behavior was observed for q = 2, where the wave function displays a ring-like shape, indicating a bind behavior of the particles. Differently from the results previously reported for the case of one particle, frozen states appear only for special combinations of the wave function parameters in case of q = 3.


Unusual diffusing regimes caused by different adsorbing surfaces

V. G. Guimaraes, H. V. Ribeiro, Q. Li, L. R. Evangelista, E. K. Lenzi, R. S. Zola
Soft Matter 11, 1658 (2015).
Anomalous Diffusion

A confined liquid with dispersed neutral particles is theoretically studied when the limiting surfaces present different dynamics for the adsorption–desorption phenomena. The investigation considers different non-singular kernels in the kinetic equations at the walls, where the suitable choice of the kernel can account for the relative importance of physisorption or chemisorption. We find that even a small difference in the adsorption–desorption rate of one surface (relative to the other) can drastically affect the behavior of the whole system. The surface and bulk densities and the dispersion are calculated when several scenarios are considered and anomalous-like behaviors are found. The approach described here is closely related to experimental situations, and can be applied in several contexts such as dielectric relaxation, diffusion-controlled relaxation in liquids, liquid crystals, and amorphous polymers.


We need more empirical investigations and model validation for a better understanding of crime - Comment on 'Statistical physics of crime: A review' by M.R. D'Orsogna and M. Perc.

H. V. Ribeiro
Physics of Life Reviews 12, 36 (2015)

Since the seminal works of Wilson and Kelling [1] in 1982, the “broken windows theory” seems to have been widely accepted among the criminologists and, in fact, empirical findings actually point out that criminals tend to return to previously visited locations. Crime has always been part of the urban society’s agenda and has also attracted the attention of scholars from social sciences ever since. Furthermore, over the past six decades the world has experienced a quick and notorious urbanization process: by the eighties the urban population was about 40% of total population, and today more than half (54%) of the world population is urban [2]. The urbanization has brought us many benefits such as better working opportunities and health care, but has also created several problems such as pollution and a considerable rise in the criminal activities. In this context of urban problems, crime deserves a special attention because there is a huge necessity of empirical and mathematical (modeling) investigations which, apart from the natural academic interest, may find direct implications for the organization of our society by improving political decisions and resource allocation.

Despite being a naturally interdisciplinary topic, the idea of a physicist studying crime may still cause some surprise (despite the fact that physicists have investigated, more than ever, several systems very far from the traditional domain of physics), but the review by D’Orsogna and Perc [3] shows us that several collective patterns related to crimes are analogous to those exhibited by classical physical systems such as the reaction–diffusion equations, which model the evolution of chemicals under chemical reactions and diffusion, but also describe the evolution of crime hotspots. D’Orsogna and Perc bring us a concise and general view of the recent applications of mathematical methods for modeling crime related problems. The review covers the modeling of crime hotspots by generalized reaction–diffusion equations and by self-exciting point process; presents an overview of the evolutionary game theory for addressing crime as a social dilemma; illustrates the use of network tools for understanding criminal organizations; and, by combining these tools with random walks methods, demonstrates how it is possible to infer the network topology of street gangs. Finally, within a more sociological view, the authors discuss the role of punishment for rehabilitation and to prevent recidivism.

What I found special in this review is that the authors do not only stay in the “physicists’ comfort zone”, that is, too focused on models and proprieties that resemble those of phase transitions, which are beautiful for theoretical physicists but much less interesting for creating methods and tools to help us prevent and understand crimes from a more social perspective. D’Orsogna and Perc drive us towards an empirical and applied approach by reviewing and discussing several sociological concepts of crime in connection with statistical models. They also present specific examples of real-world applications such as the case of the Los Angeles Police Department. This police department employed earthquake-like models aiming to prevent crime by sending police patrols to geographical areas where models indicated that crimes were more likely to occur. Another striking example is the inference of the network topology of street gangs via agent-based simulations, in which it is quite impressive to see how this simple model agrees with the empirical data.

I totally agree with the authors when they mention that methods from statistical physics can provide direct sociological implications for crime-related problems. But, in order to reach these implications, physicists need to move further away from the “comfort zone” I previously mentioned. Contrary to what occurs, for example, in materials science, there is no “social engineer” who will be responsible for implementing models and tools that we need for a better control over the criminal activities. It seems that, although this task requires skills from both science and computing (a rare combination, even today [4]), statistical physicists may represent the ideal professional to take on this challenge.


[1] J.Q. Wilson, G.L. Kelling, Broken windows, Atl Mon, 249 (1982), pp. 29–38

[2] United Nations, Department of Economic and Social Affairs, Population Division, 2014. World urbanization prospects: the 2014 revision, highlights (ST/ESA/SER.A/352).

[3] M.R. D’Orsogna, M. Perc, Statistical physics of crime: a review, Phys Life Rev (2015) [ in this issue]

[4] C.A. Mattmann, Computing: a vision for data science, Nature, 493 (2012), pp. 473–475



Investigating the interplay between mechanisms of anomalous diffusion via fractional Brownian walks on a comb-like structure

H. V. Ribeiro, A. A. Tateishi, L. G. A. Alves, R. S. Zola, E. K. Lenzi
New J. Phys. 16, 093050 (2014).
Anomalous DiffusionComb ModelComplex Systems

The comb model is a simplified description for anomalous diffusion under geometric constraints. It represents particles spreading out in a two-dimensional space where the motions in the x-direction are allowed only when the y coordinate of the particle is zero. Here, we propose an extension for the comb model via Langevin-like equations driven by fractional Gaussian noises (long-range correlated). By carrying out computer simulations, we show that the correlations in the y−direction affect the diffusive behavior in the x−direction in a non-trivial fashion, resulting in a quite rich diffusive scenario characterized by usual, superdiffusive or subdiffusive scaling of second moment in the x−direction. We further show that the long-range correlations affect the probability distribution of the particle positions in the x−direction, making their tails longer when noise in the y−direction is persistent and shorter for anti- persistent noise. Our model thus combines and allows the study/analysis of the interplay between different mechanisms of anomalous diffusion (geometric constraints and long-range correlations) and may find direct applications for describing diffusion in complex systems such as living cells.


Fractional diffusion equation, boundary conditions and surface effects

E. K. Lenzi, A. A. Tateishi, H. V. Ribeiro, M. K. Lenzi, G. Gonçalves, L. R. da Silva
J. Stat. Mech. P08019 (2014).
Anomalous Diffusion

We investigate a system governed by a fractional diffusion equation with an integro-differential boundary condition on the surface. This condition can be connected with several processes such as adsorption and/ or desorption or chemical reactions due to the presence of active sites on the surface. The solutions are obtained by using the Green function approach and show a rich class of behaviors, which can be related to anomalous diffusion.


Empirical analysis on the connection between power-law distributions and allometries for urban indicators

L. G. A. Alves, H. V. Ribeiro, E. K. Lenzi, R. S. Mendes
Physica A 409, 175 (2014).
Complex SystemsCrimeData AnalysisScaling LawsUrban Metrics

We report on the existing connection between power-law distributions and allometries. As it was first reported in Gomez-Lievano et al. (2012) for the relationship between homicides and population, when these urban indicators present asymptotic power-law distributions, they can also display specific allometries among themselves. Here, we present an extensive characterization of this connection when considering all possible pairs of relationships from twelve urban indicators of Brazilian cities (such as child labor, illiteracy, income, sanitation and unemployment). Our analysis reveals that all our urban indicators are asymptotically distributed as power laws and that the proposed connection also holds for our data when the allometric relationship displays enough correlations. We have also found that not all allometric relationships are independent and that they can be understood as a consequence of the allometric relationship between the urban indicator and the population size. We further show that the residuals fluctuations surrounding the allometries are characterized by an almost constant variance and log-normal distributions.


Reaction on a solid surface supplied by an anomalous mass transfer source

E. K. Lenzi, M. K. Lenzi, R. S. Zola, H. V. Ribeiro, F. C. Zola, L. R. Evangelista, G. Gonçalves
Physica A 410, 399 (2014).
Anomalous Diffusion

The reaction process occurring on a solid surface where active sites are present is investigated. The phenomenon is described by a linear kinetic equation capable of accounting for memory effects in the adsorption–desorption and a first order reaction process. In order to broaden the formulation of the problem, the surface is in contact with a system defined in a half space where the dynamics is governed by a fractional diffusion equation, meaning, in principle, that the approach can be applied to complex systems such as biological fluids. Our results prove that the anomalous behavior has great importance on the reaction and, consequently, on the densities rates of particles at the surface and on the distribution of particles in the bulk. The results are particularly relevant for heterogeneous catalysis.


Universal bursty behaviour in human violent conflicts

S. Picoli, M. del Castillo-Mussot, H. V. Ribeiro, E. K. Lenzi, R. S. Mendes
Sci. Rep. 4, 4773 (2014).
Complex SystemsData AnalysisWar

Understanding the mechanisms and processes underlying the dynamics of collective violence is of considerable current interest. Recent studies indicated the presence of robust patterns characterizing the size and timing of violent events in human conflicts. Since the size and timing of violent events arises as the result of a dynamical process, we explore the possibility of unifying these observations. By analyzing available catalogs on violent events in Iraq (2003–2005), Afghanistan (2008–2010) and Northern Ireland (1969–2001), we show that the inter-event time distributions (calculated for a range of minimum sizes) obeys approximately a simple scaling law which holds for more than three orders of magnitude. This robust pattern suggests a hierarchical organization in size and time providing a unified picture of the dynamics of violent conflicts.


Fractional Diffusion Equations and Equivalent Circuits Applied to Ionic Solutions

F. R. G. B. Silva, H. V. Ribeiro, M. K. Lenzi, F. S. Michels, E. K. Lenzi
Int. J. of Electrochem. Sci. 9, 1892 (2014).
Electrical Response

We investigate dilute solutions of different salts (KClO3, K2SO4, and CdCl2H2O) dissolved in Milli-Q deionized water in the context of the fractional diffusion equations and equivalent circuits. The experimental results show that in the low frequency limit the behavior of the impedance is suitable described in terms of the boundary conditons which can be connected to constant phase elements (CPE). In addition, they also indicate that salts with similar characteristics, such as the ionic potential for the negative ion, present essentially the same frequency dependence of the impedance in the low frequency limit.



Long-range spatial correlations and fluctuation statistics of lightning activity rates in Brazil

H. V. Ribeiro, F. J. Antonio, L. G. A. Alves, E. K. Lenzi, R. S. Mendes
EPL 104, 69001 (2013).
Complex SystemsData AnalysisLightning

We report on a statistical analysis of the lightning activity rates in all Brazilian cities. We find out that the average of lightning activity rates exhibits a dependence on the latitude of the cities, displaying one peak around the Tropic of Capricorn and another one just before the Equator. We verify that the standard deviation of these rates is almost a constant function of the latitude and that the distribution of the fluctuations surrounding the average tendency is quite well described by a Gumbel distribution, which thus connects these rates to extreme processes. We also investigate the behavior of the lightning activity rates vs. the longitude of the cities. For this case, the average rates exhibit an approximate plateau for a wide range of longitude values, the standard deviation is an approximate constant function of longitude, and the fluctuations are described by a Laplace distribution. We further characterize the spatial correlation of the lightning activity rates between pairs of cities, where our results show that the spatial correlation function decays very slowly with the distance between the cities and that for intermediate distances the correlation exhibits an approximate logarithmic decay. Finally, we propose to model this last behavior within the framework of the Edwards-Wilkinson equation.


First passage time for a diffusive process under a geometric constraint

A. A. Tateishi, F. S. Michels, M. A. F. Santos, E. K. Lenzi, H. V. Ribeiro
J. Stat. Mech. P09017 (2013).
Anomalous DiffusionComb Model

We investigate the solutions, survival probability, and first passage time for a two-dimensional diffusive process subjected to the geometric constraints of a backbone structure. We consider this process governed by a fractional Fokker–Planck equation by taking into account the boundary conditions ρ(0, y; t) = ρ(∞, y; t) = 0, ρ(x, ±∞; t) = 0, and an arbitrary initial condition. Our results show an anomalous spreading and, consequently, a nonusual behavior for the survival probability and for the first passage time distribution that may be characterized by different regimes. In addition, depending on the choice of the parameters present in the fractional Fokker–Planck equation, the survival probability indicates that part of the system may be trapped in the branches of the backbone structure.


Engagement in the electoral processes: Scaling laws and the role of the political positions

M. C. Mantovani, H. V. Ribeiro, E. K. Lenzi, S. Picoli Jr., R. S. Mendes
Phys. Rev. E 88, 024802 (2013).
Complex SystemsData AnalysisElectionsScaling Laws

We report on a statistical analysis of the engagement in the electoral processes of all Brazilian cities by considering the number of party memberships and the number of candidates for mayor and councillor. By investigating the relationships between the number of party members and the population of voters, we have found that the functional forms of these relationships are well described by sublinear power laws (allometric scaling) surrounded by a multiplicative log-normal noise. We have observed that this pattern is quite similar to those we previously reported for the relationships between the number of candidates (mayor and councillor) and population of voters [Europhys. Lett. 96, 48001 (2011)], suggesting that similar universal laws may be ruling the engagement in the electoral processes. We also note that the power-law exponents display a clear hierarchy, where the more influential is the political position the smaller is the value of the exponent. We have also investigated the probability distributions of the number of candidates (mayor and councillor), party memberships, and voters. The results indicate that the most influential positions are characterized by distributions with very short tails, while less influential positions display an intermediate power-law decay before showing an exponential-like cutoff. We discuss the possibility that, in addition to the political power of the position, limitations in the number of available seats can also be connected with this changing of behavior. We further believe that our empirical findings point out to an under-representation effect, where the larger the city is, the larger are the obstacles for more individuals to become directly engaged in the electoral process.


Distance to the scaling law: a useful approach for unveiling relationships between crime and urban metrics

L. G. A. Alves, H. V. Ribeiro, E. K. Lenzi, R. S. Mendes
PLoS One 8, e69580 (2013).
Complex SystemsCrimeData AnalysisUrban Metrics

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 i) the non-explanatory potential of the elderly population when the number of homicides is much above or much below the scaling law, ii) the fact that unemployment has explanatory potential only when the number of homicides is considerably larger than the expected by the power law, and iii) 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.


Time dependent solutions for a fractional Schrödinger equation with delta potentials

E. K. Lenzi, H. V. Ribeiro, M. A. F. Santos, R. Rossato, R. S. Mendes
J. Math Phys. 54, 082107 (2013).
Fractional Schrödinger

We investigate, for an arbitrary initial condition, the time dependent solutions for a fractional Schro ̈dinger equation in the presence of delta potentials by using the Green function approach. The solutions obtained show an anomalous spreading asymptotically characterized by a power-law behavior, which is governed by the order of the fractional spatial operator present in the Schro ̈dinger equation.


Anti-persistent behavior of defects in a lyotropic liquid crystal during annihilation

H. V. Ribeiro, R. R. Guimaraes, R. T. Teixeira-Souza, H. Mukai, P. R. G. Fernandes, R. S. Mendes
Phys. Rev. E 87, 054501 (2013).
Complex SystemsData AnalysisLiquid Crystal

We report on the dynamical behavior of defects of strength s = ±1/2 in a lyotropic liquid crystal during the annihilation process. By following their positions using time-resolved polarizing microscopy technique, we present statistically significant evidence that the relative velocity between defect pairs is Gaussian distributed, antipersistent, and long-range correlated. We further show that simulations of the Lebwohl-Lasher model reproduce quite well our experimental findings.


Scaling laws in the dynamics of crime growth rate

L. G. A. Alves, H. V. Ribeiro, R. S. Mendes
Physica A 392, 2676 (2013).
Complex SystemsCrimeData AnalysisScaling Laws

The increasing number of crimes in areas with large concentrations of people have made cities one of the main sources of violence. Understanding characteristics of how crime rate expands and its relations with the cities size goes beyond an academic question, being a central issue for contemporary society. Here, we characterize and analyze quantitative aspects of murders in the period from 1980 to 2009 in Brazilian cities. We find that the distribution of the annual, biannual and triannual logarithmic homicide growth rates exhibit the same functional form for distinct scales, that is, a scale invariant behavior. We also identify asymptotic power-law decay relations between the standard deviations of these three growth rates and the initial size. Further, we discuss similarities with complex organizations.


Move-by-move dynamics of the advantage in chess matches reveals population-level learning of the game

H. V. Ribeiro, R. S. Mendes, E. K. Lenzi, M. del Castillo-Mussot, L. A. N. Amaral
PLoS One 8, e54165 (2013).
ChessComplex SystemsData AnalysisLearning

The complexity of chess matches has attracted broad interest since its invention. This complexity and the availability of large number of recorded matches make chess an ideal model systems for the study of population-level learning of a complex system. We systematically investigate the move-by-move dynamics of the white player’s advantage from over seventy thousand high level chess matches spanning over 150 years. We find that the average advantage of the white player is positive and that it has been increasing over time. Currently, the average advantage of the white player is~0.17 pawns but it is exponentially approaching a value of 0.23 pawns with a characteristic time scale of 67 years. We also study the diffusion of the move dependence of the white player’s advantage and find that it is non-Gaussian, has long-ranged anti-correlations and that after an initial period with no diffusion it becomes super-diffusive. We find that the duration of the non-diffusive period, corresponding to the opening stage of a match, is increasing in length and exponentially approaching a value of 15.6 moves with a characteristic time scale of 130 years. We interpret these two trends as a resulting from learning of the features of the game. Additionally, we find that the exponent a characterizing the super-diffusive regime is increasing toward a value of 1.9, close to the ballistic regime. We suggest that this trend is due to the increased broadening of the range of abilities of chess players participating in major tournaments.


Diffusive process on a backbone structure with drift terms

E. K. Lenzi, L. R. da Silva, A. A. Tateishi, M. K. Lenzi, H. V. Ribeiro
Phys. Rev. E 87, 012121 (2013).
Anomalous Diffusion

The effects of an external force on a diffusive process subjected to a backbone structure are investigated by considering the system governed by a Fokker-Planck equation with drift terms. Our results show an anomalous spreading which may present different diffusive regimes connected to anomalous diffusion and stationary states.


Anomalous Diffusion and Electrical Response of Ionic Solutions

E. K. Lenzi, P. R. G. Fernandes, T. Petrucci, H. Mukai, H. V. Ribeiro, M. K. Lenzi, G. G. Lenzi
Int. J. of Electrochem. Sci. 8, 2849 (2013).
Electrical Response

We analyze the electrical response obtained in the framework of a model in which the diffusion of mobile ions in the bulk is governed by a fractional diffusion equation of distributed order subjected to integro-differential boundary conditions. The analysis is carried out by supposing that the positive and negative ions have different mobility and that the electric potential profile across the sample satisfies the Poisson’s equation. In addition, we also compare the analytical results with experimental data obtained from ionic solutions of a salt dissolved in water, reveling a good agreement and evidencing that the dynamics of the ions can be related to different diffusive processes and, consequently, to anomalous diffusion.



Anomalous diffusion and long-range correlations in the score evolution of the game of cricket

H. V. Ribeiro, S. Mukherjee, X. H. T. Zeng,
Phys. Rev. E 86, 022102 (2012).
Complex SystemsCricketData AnalysisHot Hands

We investigate the time evolution of the scores of the second most popular sport in the world: the game of cricket. By analyzing, event by event, the scores of more than 2000 matches, we point out that the score dynamics is an anomalous diffusive process. Our analysis reveals that the variance of the process is described by a power-law dependence with a superdiffusive exponent, that the scores are statistically self-similar following a universal Gaussian distribution, and that there are long-range correlations in the score evolution. We employ a generalized Langevin equation with a power-law correlated noise that describes all the empirical findings very well. These observations suggest that competition among agents may be a mechanism leading to anomalous diffusion and long-range correlation.


Fractional Schrödinger equation with noninteger dimensions

J. Martins, H. V. Ribeiro, L. R. Evangelista, L. R. da Silva, E. K. Lenzi
Applied Mathematics and Computation 219, 2313 (2012).
Fractional Schrödinger

The spatial and time dependent solutions of the Schrödinger equation incorporating the fractional time derivative of distributed order and extending the spatial operator to nonin- teger dimensions are investigated. They are obtained by using the Green function approach in two situations: the free case and in the presence of a harmonic potential. The results obtained show an anomalous spreading of the wave packet which may be related to an anomalous diffusion process.


Complexity-Entropy Causality Plane as a Complexity Measure for Two-dimensional Patterns

H. V. Ribeiro, L. Zunino, E. K. Lenzi, P. A. Santoro, R. S. Mendes
PLoS One 7, e40689 (2012).
Complexity MeasureEntropyTwo-dimensional Patterns

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 i) fractal landscapes generated numerically where we compare our measures with the Hurst exponent; ii) liquid crystal textures where nematic-isotropic-nematic phase transitions were properly identified; iii) 12 characteristic textures of liquid crystals where the different values show that the method can distinguish different phases; iv) and Ising surfaces where our method identified the critical temperature and also proved to be stable.


Different diffusive regimes, generalized Langevin and diffusion equations

A. A. Tateishi, E. K. Lenzi, L. R. da Silva, H. V. Ribeiro, S. Picoli Jr., R. S. Mendes
Phys. Rev. E 85, 011147 (2012).
Anomalous DiffusionLangevin Equation

We investigate a generalized Langevin equation (GLE) in the presence of an additive noise characterized by the mixture of the usual white noise and an arbitrary one. This scenario lead us to a wide class of diffusive processes, in particular the ones whose noise correlation functions are governed by power laws, exponentials, and Mittag-Leffler functions. The results show the presence of different diffusive regimes related to the spreading of the system. In addition, we obtain a fractional diffusionlike equation from the GLE, confirming the results for long time.


Solution for fractional diffusion equation with noninteger dimensions

L. S. Lucena, L. R. da Silva, A. A. Tateishi, H. V. Ribeiro, E. K. Lenzi
Nonlinear Anal.: Real World Appl. 13, 1955 (2012).
Anomalous Diffusion

We investigate a fractional diffusion equation with a nonlocal reaction term by using the Green function approach. We also consider a modified spatial operator in order to cover situations characterized by a noninteger dimension. The results show a nonusual spreading of the initial condition which can be connected to a rich class of anomalous diffusive processes.


Complexity-entropy causality plane: a useful approach for distinguishing songs

H. V. Ribeiro, L. Zunino, R. S. Mendes, E. K. Lenzi
Physica A 391, 2421 (2012).
Complex SystemsData AnalysisEntropyMusic

Nowadays we are often faced with huge databases resulting from the rapid growth of data storage technologies. This is particularly true when dealing with music databases. In this context, it is essential to have techniques and tools able to discriminate properties from these massive sets. In this work, we report on a statistical analysis of more than ten thousand songs aiming to obtain a complexity hierarchy. Our approach is based on the estimation of the permutation entropy combined with an intensive complexity measure, building up the complexity–entropy causality plane. The results obtained indicate that this representation space is very promising to discriminate songs as well as to allow a relative quantitative comparison among songs. Additionally, we believe that the here-reported method may be applied in practical situations since it is simple, robust and has a fast numerical implementation.


Continuous Time Random Walk and different diffusive regimes

H. V. Ribeiro, R. Rossato, A. A. Tateishi, E. K. Lenzi, R. S. Mendes
Acta Sci.-Technol. 34, 201 (2012).
Anomalous DiffusionCTRW

We investigate how it is possible to obtain different diffusive regimes from the Continuous Time Random Walk approach performing suitable changes for the waiting time and jumping distributions in order to get two or more regimes for the same diffusive process. We also obtain diffusion- like equations related to these processes and investigate the connection of the results with anomalous diffusion.



Anomalous-diffusion approach applied to the electrical response of water

E. K. Lenzi, P. R. G. Fernandes, T. Petrucci, H. Mukai, H. V. Ribeiro
Phys. Rev. E 84, 041128 (2011).
Anomalous DiffusionElectrical Response

We investigate the electrical response of Milli-Q deionized water by using a fractional diffusion equation of distributed order with the interfaces (i.e., the boundary conditions at the electrodes limiting the sample) governed by integro-differential equations. We also consider that the positive and negative ions have the same mobility and that the electric potential profile across the sample satisfies Poisson’s equation. In addition, the good agreement between the experimental data and this approach evidences the presence of anomalous diffusion due to the surface effects in this system.


Scaling laws and universality in the choice of election candidates

M. C. Mantovani, H. V. Ribeiro, M. V. Moro, S. Picoli Jr., R. S. Mendes
EPL 96, 48001 (2011).
Complex SystemsData AnalysisElectionsScaling Laws

Nowadays there is an increasing interest of physicists in finding regularities related to social phenomena. This interest is clearly motivated by applications that a statistical mechanical description of the human behavior may have in our society. By using this framework, we address this work to cover an open question related to elections: the choice of elections candidates (candidature process). Our analysis reveals that, apart from the social motivations, this system displays features of traditional out-of-equilibrium physical phenomena such as scale-free statistics and universality. Basically, we found a non-linear (power law) mean correspondence between the number of candidates and the size of the electorate (number of voters), and also that this choice has a multiplicative underlying process (lognormal behavior). The universality of our findings is supported by data from 16 elections from 5 countries. In addition, we show that aspects of scale-free network can be connected to this universal behavior.


Exact propagator for a Fokker-Planck equation, first passage time distribution, and anomalous diffusion

A. T. Silva, E. K. Lenzi, L. R. Evangelista, M. K. Lenzi, H. V. Ribeiro, A. A. Tateishi
J. Math Phys. 52, 083301 (2011).
Anomalous Diffusion

We obtain an exact form for the propagator of the Fokker-Planck equation ∂tρ = ∂x (D(x)∂x ρ) −∂x (F(x, t)ρ), with D(x) = D|x|−η in presence of the external force F(x, t) = −k(t)x + (K/x) |x|−η. Using the results found here, we also investigate the mean square displacement, survival probability, and first passage time distribution. In addition, we discuss the connection of these results with anomalous diffusion phenomena.


Non-Markovian diffusion equation and diffusion in a porous catalyst

E. K. Lenzi, H. V. Ribeiro, J. Martins, M. K. Lenzi, G. G. Lenzi, S. Spheccia
Chem. Eng. J. 172, 1083 (2011).
Anomalous Diffusion

We revisit the problem of diffusion in a porous catalyst by incorporating in the diffusion equation fractional time derivatives and a spatial dependent diffusion coefficient in order to extend the usual description to situations which have an unusual behavior. In our analysis, we also consider a nonlocal reaction term of linear order. We obtain exact solutions for the profile of substance in the porous catalyst in terms of the Green function approach. The results show an anomalous behavior of the concentration profile spreading which may be connected to anomalous diffusion.


Anomalous diffusion in a symbolic model

H. V. Ribeiro, E. K. Lenzi, R. S. Mendes, P. A. Santoro
Physica Scripta 83, 045007 (2011).
Anomalous DiffusionCTRWSymbolic Dynamics

In this work, we investigate some statistical properties of symbolic sequences generated by a numerical procedure in which the symbols are repeated following the power-law probability density. In this analysis, we consider that the sum of n symbols represents the position of a particle in erratic movement. This approach reveals a rich diffusive scenario characterized by non-Gaussian distribution and, depending on the power-law exponent or the procedure used to build the walker, we may have superdiffusion, subdiffusion or usual diffusion. Additionally, we use the continuous-time random walk framework to compare the analytic results with the numerical data, thereby finding good agreement. Because of its simplicity and flexibility, this model can be a candidate for describing real systems governed by power-law probability densities.


Spreading patterns of the influenza A (H1N1) pandemic

S. Picoli Jr., J. J. V. Teixeira, H. V. Ribeiro, L. C. Malacarne, R. P. B. Santos, R. S. Mendes
PLoS One 6, e17823 (2011).
Complex SystemsData AnalysisInfluenza A

We investigate the dynamics of the 2009 influenza A (H1N1/S-OIV) pandemic by analyzing data obtained from World Health Organization containing the total number of laboratory-confirmed cases of infections – by country – in a period of 69 days, from 26 April to 3 July, 2009. Specifically, we find evidence of exponential growth in the total number of confirmed cases and linear growth in the number of countries with confirmed cases. We also find that, i) at early stages, the cumulative distribution of cases among countries exhibits linear behavior on log-log scale, being well approximated by a power law decay; ii) for larger times, the cumulative distribution presents a systematic curvature on log-log scale, indicating a gradual change to lognormal behavior. Finally, we compare these empirical findings with the predictions of a simple stochastic model. Our results could help to select more realistic models of the dynamics of influenza-type pandemics.


On the dynamics of bubbles in boiling water

H. V. Ribeiro, R. S Mendes, E. K. Lenzi, M. P. Belancon, L. C. Malacarne
Chaos Solitons and Fractals 44, 178 (2011).
Boiling WaterComplex Systems

We investigate the dynamics of many interacting bubbles in boiling water by using a laser scattering experiment. Specifically, we analyze the temporal variations of a laser intensity signal which passed through a sample of boiling water. Our empirical results indicate that the return interval distribution of the laser signal does not follow an exponential distribution; contrariwise, a heavy-tailed distribution has been found. Additionally, we compare the experimental results with those obtained from a minimalist phenomenological model, finding a good agreement.


Solutions for a diffusion equation with a backbone term

A. A. Tateishi, E. K. Lenzi, H. V. Ribeiro, L. R. Evangelista, R. S. Mendes
J. Stat. Mech. P02022 (2011).
Anomalous DiffusionComb Model

We investigate the diffusion equation ∂t ρ = Dy ∂y2 ρ + Dx ∂x2 ρ + D ̄xδ(y)∂xμρ subjected to the boundary conditions ρ(±∞, y; t) = 0 and ρ(x,±∞;t) = 0, and the initial condition ρ(x,y;0) = ρˆ(x,y). We obtain exact solutions in terms of the Green function approach and analyze the mean square displacement in the x and y directions. This analysis shows an anomalous spreading of the system which is characterized by different diffusive regimes connected to anomalous diffusion.


The soundscape dynamics of human agglomeration

H. V. Ribeiro, R. T. de Souza, E. K. Lenzi, R. S. Mendes, L. R. Evangelista
New J. Phys. 13, 023028 (2011).
Complex SystemsData AnalysisHuman Dynamics

We report on a statistical analysis of the people agglomeration soundscape. Specifically, we investigate the normalized sound amplitudes and intensities that emerge from human collective meetings. Our findings support the existence of non-trivial dynamics characterized by heavy tail distributions in the sound amplitudes, long-range correlations in the sound intensity and non-exponential distributions in the return interval distributions. Additionally, motivated by the time-dependent behavior present in the volatility/variance series, we compare the observational data with those obtained from a minimalist autoregressive stochastic model, namely the generalized autoregressive conditional heteroskedastic process (the GARCH process), and find that there is good agreement.


Universal patterns in sound amplitudes of songs and music genres

R. S. Mendes, H. V. Ribeiro, F. C. M. Freire, A. A. Tateishi, E. K. Lenzi
Phys. Rev. E 83, 017101 (2011).
Complex SystemsData AnalysisMusic

We report a statistical analysis of more than eight thousand songs. Specifically, we investigated the probability distribution of the normalized sound amplitudes. Our findings suggest a universal form of distribution that agrees well with a one-parameter stretched Gaussian. We also argue that this parameter can give information on music complexity, and consequently it helps classify songs as well as music genres. Additionally, we present statistical evidence that correlation aspects of the songs are directly related to the non-Gaussian nature of their sound amplitude distributions.



Earthquake-like patterns of acoustic emission in crumpled plastic sheets

R. S. Mendes, L. C. Malacarne, R. P. B. Santos, H. V. Ribeiro, S. Picoli Jr.
EPL 92, 29001 (2010).
Data AnalysisEarthquakesPlastic Sheets

We report remarkable similarities in the output signal of two distinct out-of-equilibrium physical systems – earthquakes and the intermittent acoustic noise emitted by crumpled plastic sheets, i.e. Biaxially Oriented Polypropylene (BOPP) films. We show that both signals share several statistical properties including the distribution of energy, distribution of energy increments for distinct time scales, distribution of return intervals and correlations in the magnitude and sign of energy increments. This analogy is consistent with the concept of universality in complex systems and could provide some insight on the mechanisms behind the complex behavior of earthquakes.


Continuous-time random walk as a guide to fractional Schrödinger equation

E. K. Lenzi, H. V. Ribeiro, H. Mukai, R. S. Mendes
J. Math Phys. 51, 092102 (2010).
CTRWFractional Schrödinger

We argue that the continuous-time random walk approach may be a useful guide to extend the Schrödinger equation in order to incorporate nonlocal effects, avoiding the inconsistencies raised by Jeng et al. [J. Math. Phys. 51, 062102 (2010)]. As an application, we work out a free particle in a half space, obtaining the time depen- dent solution by considering an arbitrary initial condition.


Solutions for a non-Markovian diffusion equation

E. K. Lenzi, L. R. Evangelista, M. K. Lenzi, H. V. Ribeiro, E.C. de Oliveira
Phys. Lett. A 374, 4193 (2010).
Anomalous Diffusion

Solutions for a non-Markovian diffusion equation are investigated. For this equation, we consider a spatial and time dependent diffusion coefficient and the presence of an absorbent term. The solutions exhibit an anomalous behavior which may be related to the solutions of fractional diffusion equations and anomalous diffusion.


Dynamics of tournaments: the soccer case

H. V. Ribeiro, R. S. Mendes, L. C. Malacarne, S. Picoli Jr., P. A. Santoro
Eur. Phys. J. B 75, 327 (2010).
Complex SystemsData AnalysisSoccer

A random walk-like model is considered to discuss statistical aspects of tournaments. The model is applied to soccer leagues with emphasis on the scores. This competitive system was computationally simulated and the results are compared with empirical data from the English, the German and the Spanish leagues and showed a good agreement with them. The present approach enabled us to characterize a diffusion where the scores are not normally distributed, having a short and asymmetric tail extending towards more positive values. We argue that this non-Gaussian behavior is related with the difference between the teams and with the asymmetry of the scores system. In addition, we compared two tournament systems: the all-play-all and the elimination tournaments.



Symbolic Sequences and Tsallis Entropy

H. V. Ribeiro, E. K. Lenzi, R. S. Mendes, G. A. Mendes, L. R. da Silva
Braz. J. Phys. 39, 444 (2009).

We address this work to investigate symbolic sequences with long-range correlations by using computational simulation. We analyze sequences with two, three and four symbols that could be repeated l times, with the probability distribution p(l) ∝ 1/lμ. For these sequences, we verified that the usual entropy increases more slowly when the symbols are correlated and the Tsallis entropy exhibits, for a suitable choice of q, a linear behavior. We also study the chain as a random walk-like process and observe a nonusual diffusive behavior depending on the values of the parameter μ.