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Permutation entropy and its associated frameworks are remarkable examples of physics-inspired techniques adept at processing complex and extensive datasets. Despite substantial progress in developing and applying these tools, their use has been predominantly limited to structured datasets such as time series or images. Here, we introduce the k-nearest neighbor permutation entropy, an innovative extension of the permutation entropy tailored for unstructured data, irrespective of their spatial or temporal configuration and dimensionality. Our approach builds upon nearest neighbor graphs to establish neighborhood relations and uses random walks to extract ordinal patterns and their distribution, thereby defining the k-nearest neighbor permutation entropy. This tool not only adeptly identifies variations in patterns of unstructured data, but also does so with a precision that significantly surpasses conventional measures such as spatial autocorrelation. Additionally, it provides a natural approach for incorporating amplitude information and time gaps when analyzing time series or images, thus significantly enhancing its noise resilience and predictive capabilities compared to the usual permutation entropy. Our research substantially expands the applicability of ordinal methods to more general data types, opening promising research avenues for extending the permutation entropy toolkit for unstructured data., Comment: 14 two-column pages, 6 figures

Power law scaling models have been used to understand the complexity of systems as diverse as cities, neurological activity, and rainfall and lightning. In the scaling framework, power laws and standard linear regression methods are widely used to estimate model parameters with assumed normality and fixed variance. Generalized linear models (GLM) can accommodate a wider range of distributions where the chosen distribution must meet the assumptions of the data to prevent model bias. We present a widely applicable Bayesian generalized logistic regression (BGLR) framework to more flexibly model a continuous real response addressing skew and heteroscedasticity. The Generalized Logistic Distribution (GLD) was selected to flexibly model skewed continuous data. This resulted in a nonlinear posterior distribution which may not have an analytical solution which can be solved numerically with Markov Chain Monte Carlo (MCMC) methods. We compared the BGLR model to standard and Bayesian normal models having fixed and varying variance when fitting power laws to 759 days of COVID-19 data. The BGLR yielded information beyond existing methods about the evolution of skew and skedasticity while revealing parameter bias of widely used methods. The BGLR flexibly modelled the complex characteristics necessary for an improved understanding of the propagation and dynamics of this infectious disease. The model is generally applicable and can be used as a template for modeling complexity with other distributions., Comment: Manuscript and supplementary combined has 35 pages. The manuscript has 8 figures and the supplement has 1 figure

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., Comment: 11 two-column pages, 2 figures, supplementary information; accepted for publication in Physical Review Research

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., Comment: 13 two-column pages, 6 figures; accepted for publication in Communications Physics

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 Maring\'a, 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. Maring\'a 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., Comment: 20 pages, 5 figures, supplementary information; accepted for publication in Scientific Reports

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., Comment: 14 two-column pages, 5 figures

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., Comment: 16 pages, 4 figures, supplementary information; accepted for publication in Scientific Reports

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., Comment: 16 pages, 4 figures, supplementary information; accepted for publication in Scientific Reports

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., Comment: 10 pages, 4 figures, supplementary information; accepted for publication in Scientific Reports

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., Comment: 6 two-column pages, 1 figure, supplementary information; accepted for publication in EPL

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., Comment: 11 pages, 6 figures, supplementary information; accepted for publication in Scientific Reports

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., Comment: 23 two-column pages, 30 figures, supplementary information; accepted for publication in Physical Review E

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., Comment: 9 two-column pages, 4 figures; accepted for publication in Chaos, Solitons & Fractals

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 threshold., Comment: 11 pages, 4 figures, supplementary information; accepted for publication in Scientific Reports

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 non-outlier performance. Outlier researchers either outperform in productivity or journal prestige, but they rarely do so in both categories. Non-outliers 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., Comment: 14 two-column pages, 4 figures, supplementary information; accepted for publication in Physical Review Research

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 triggered increased variance indicative of outbreaks with local impact and country scale heterogeneity. University reopening and December holidays triggered 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 to 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., Comment: Main paper has 7 Figures. The full document also contains a supplement with an additional 8 figures. These 8 supplementary figures contain multiple (300+) panels representing each day of the study period

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 the 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 R\'enyi 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., Comment: 23 two-column pages, 10 figures; accepted for publication in Chaos

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., Comment: 17 pages, 3 figures, supplementary information; accepted for publication in PLoS ONE

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., Comment: 11 pages, 4 figures, supplementary information; accepted for publication in Scientific Reports

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., Comment: 15 two-column pages, 6 figures; accepted for publication in Physical Review E

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., Comment: 8 two-column pages, 4 figures, supplementary information

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., Comment: 11 pages, 4 figures, supplementary information; accepted for publication in Scientific Reports

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 its 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 non-random 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 seismic activity caused by large earthquakes., Comment: 14 two-column pages, 8 figures; accepted for publication in Physical Review E

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., Comment: 13 two-column pages, 2 figures, supplementary information; accepted for publication in Nature Communications

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., Comment: 10 pages, 4 figures; accepted for publication in Scientific Reports

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., Comment: 19 pages, 4 figures, 1 table, supplementary information; accepted for publication in Physica A

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 thousand 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 that the employed complexity-entropy measures can be used for an effective classification of artworks., Comment: 10 two-column pages, 5 figures; accepted for publication in PNAS [supplementary information available at http://www.pnas.org/highwire/filestream/824089/field_highwire_adjunct_files/0/pnas.1800083115.sapp.pdf]

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., Comment: 28 pages, 6 figures, 2 tables; accepted for publication in PLoS ONE

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 R\'enyi entropy, which is another monoparametric generalization of the Shannon entropy. By combining the R\'enyi entropy with the proper generalization of the statistical complexity, we associate a parametric curve (the R\'enyi 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., Comment: 19 pages, 6 figures; accepted for publication in Physica A

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 in Brazil 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., Comment: 20 pages, 5 figures; accepted for publication in Journal of Complex Networks

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., Comment: Accepted for publication in Physica A

Unveiling the relationships between crime and socioeconomic factors is crucial for modeling and preventing these illegal activities. Recently, a significant advance has been made in understanding the influence of urban metrics on the levels of crime in different urban systems. In this chapter, we show how the dynamics of crime growth rate and the number of crime in cities are related to cities' size. We also discuss the role of urban metrics in crime modeling within the framework of the urban scaling hypothesis, where a data-driven approach is proposed for modeling crime. This model provides several insights into the mechanism ruling the dynamics of crime and can assist policymakers in making better decisions on resource allocation and help crime prevention., Comment: Book Chapter in 'Understanding Crime Through Science - Interdisciplinary Analysis and Modeling of Criminal Activities' - Springer, 2018

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 show that these new operators are a simple and efficient way for incorporating different structural aspects into the system, opening new possibilities for modeling and investigating anomalous diffusive processes.

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 mono-parametric 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/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., Comment: Accepted for publication in PRE

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., Comment: Accepted for publication in Chaos, Solitons & Fractals

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., Comment: Accepted for publication in EPL

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 every 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., Comment: Accepted for publication in PLoS ONE

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., Comment: Accepted for publication in PLoS ONE

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., 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., Comment: Accepted for publication in PLoS ONE

We report on the time dependent solutions of the $q-$generalized Schr\"odinger equation proposed by Nobre et al. [Phys. Rev. Lett. 106, 140601 (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 Schr\"odinger equation is recovered in the limit of $q\rightarrow 1$. 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 $q=3$., Comment: Accepted for publication in Physica A

We report on the existing connection between power-law distributions and allometries. As it was first reported in [PLoS ONE 7, e40393 (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., Comment: Accepted for publication in Physica A

We report on a quantitative analysis of relationships between the number of homicides, population size and other ten 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 have 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., Comment: Accepted for publication in PLoS ONE