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

In his pioneering research, G. K. Zipf formulated a couple of statistical laws on the relationship between the frequency of a word with its number of meanings: the law of meaning distribution, relating the frequency of a word and its frequency rank, and the meaning-frequency law, relating the frequency of a word with its number of meanings. Although these laws were formulated more than half a century ago, they have been only investigated in a few languages. Here we present the first study of these laws in Catalan. We verify these laws in Catalan via the relationship among their exponents and that of the rank-frequency law. We present a new protocol for the analysis of these Zipfian laws that can be extended to other languages. We report the first evidence of two marked regimes for these laws in written language and speech, paralleling the two regimes in Zipf's rank-frequency law in large multi-author corpora discovered in early 2000s. Finally, the implications of these two regimes will be discussed., Comment: 21 pages, 11 figures

This paper examines the impact of widespread adoption of information and communication technologies (ICT) on urban structure worldwide. Has it offset agglomeration benefits and led to more dispersed spatial structures, or has it strengthened urban externalities and thus resulted in more concentrated spatial structures? Theoretical and empirical studies on this question have produced contradictory findings. The present study recognizes that assumptions made earlier about the evolution of technological capabilities do not necessarily hold today. As cutting-edge digital technologies have matured considerably, a fresh look at this question is called for. The paper addresses this issue by means of several data sets using instrumental variable methods. One is the UN data on Urban Settlements with more than 300, 000 inhabitants. Estimation methods with these data show that increased adoption of ICT has resulted in national urban systems that are less uniform in terms of city sizes and are characterized by higher population concentrations in larger cities, when concentration is proxied the Pareto (Zipf) coefficient for national city size distributions. Two, is disaggregated data for the urban systems of the US, defined as Micropolitan and Metropolitan Areas, and for the UK, defined as Built-up Areas in England and Wales, respectively. These data allow for the impacts to be studied for cities smaller than those included in the cross-country data. Increased internet usage improved a city’s ranking in the US urban system. Similarly, increased download speed improves a built-up area’s ranking in England and Wales.

In this article, we study the applicability of Benford’s law and Zipf’s law to national COVID-19 case figures with the aim of establishing guidelines upon which methods of fraud detection in epidemiology, based on formal statistical analysis, can be developed. Moreover, these approaches may also be used in evaluating the performance of public health surveillance systems. We provide theoretical arguments for why the empirical laws should hold in the early stages of an epidemic, along with preliminary empirical evidence in support of these claims. Based on data published by the World Health Organization and various national governments, we find empirical evidence that suggests that both Benford’s law and Zipf’s law largely hold across countries, and deviations can be readily explained. To the best of our knowledge, this paper is among the first to present a practical application of Zipf’s law to fraud detection.

Normalization is crucial in RNA-seq data analyses. Due to the existence of excessive zeros and a large number of small measures, it is challenging to find reliable linear rescaling normalization parameters. We propose a Zipf plot based normalization method (ZN) assuming that all gene profiles have similar upper tail behaviors in their expression distributions. The new normalization method uses global information of all genes in the same profile without gene-level expression alteration. It doesn't require the majority of genes to be not differentially expressed (DE), and can be applied to data where the majority of genes are weakly or not expressed. Two normalization schemes are implemented with ZN: a linear rescaling scheme and a non-linear transformation scheme. The linear rescaling scheme can be applied alone or together with the non-linear normalization scheme. The performance of ZN is benchmarked against five popular linear normalization methods for RNA-seq data. Results show that the linear rescaling normalization scheme by itself works well and is robust. The non-linear normalization scheme can further improve the normalization outcomes and is optional if the Zipf plots show parallel patterns.

We present an equivalence between stochastic and deterministic variable approaches to represent ranked data and find the expressions obtained to be suggestive of statistical-mechanical meanings. We first reproduce size-rank distributions N(k) from real data sets by straightforward considerations based on the assumed knowledge of the background probability distribution P(N) that generates samples of random variable values similar to real data. The choice of different functional expressions for P(N): power law, exponential, Gaussian, etc., leads to different classes of distributions N(k) for which we find examples in nature. Then we show that all of these types of functions can be alternatively obtained from deterministic dynamical systems. These correspond to one-dimensional nonlinear iterated maps near a tangent bifurcation whose trajectories are proved to be precise analogues of the N(k). We provide explicit expressions for the maps and their trajectories and find they operate under conditions of vanishing or small Lyapunov exponent, therefore at or near a transition to or out of chaos. We give explicit examples ranging from exponential to logarithmic behavior, including Zipf’s law. Adoption of the nonlinear map as the formalism central character is a useful viewpoint, as variation of its few parameters, that modify its tangency property, translate into the different classes for N(k).

In this study, we compare statistical properties of ancient and modern Chinese within the framework of weighted complex networks. We examine two language networks based on different Chinese versions of the Records of the Grand Historian. The comparative results show that Zipf’s law holds and that both networks are scale-free and disassortative. The interactivity and connectivity of the two networks lead us to expect that the modern Chinese text would have more phrases than the ancient Chinese one. Furthermore, by considering some of the topological and weighted quantities, we find that expressions in ancient Chinese are briefer than in modern Chinese. These observations indicate that the two languages might have different linguistic mechanisms and combinatorial natures, which we attribute to the stylistic differences and evolution of written Chinese.

The performance of deep learning in natural language processing has been spectacular, but the reasons for this success remain unclear because of the inherent complexity of deep learning. This paper provides empirical evidence of its effectiveness and of a limitation of neural networks for language engineering. Precisely, we demonstrate that a neural language model based on long short-term memory (LSTM) effectively reproduces Zipf's law and Heaps' law, two representative statistical properties underlying natural language. We discuss the quality of reproducibility and the emergence of Zipf's law and Heaps' law as training progresses. We also point out that the neural language model has a limitation in reproducing long-range correlation, another statistical property of natural language. This understanding could provide a direction for improving the architectures of neural networks., Comment: 21 pages, 11 figures

We examine the relationship between two different types of ranked data, frequencies and magnitudes. We consider data that can be sorted out either way, through numbers of occurrences or size of the measures, as it is the case, say, of moon craters, earthquakes, billionaires, etc. We indicate that these two types of distributions are functional inverses of each other, and specify this link, first in terms of the assumed parent probability distribution that generates the data samples, and then in terms of an analog (deterministic) nonlinear iterated map that reproduces them. For the particular case of hyperbolic decay with rank the distributions are identical, that is, the classical Zipf plot, a pure power law. But their difference is largest when one displays logarithmic decay and its counterpart shows the inverse exponential decay, as it is the case of Benford law, or viceversa. For all intermediate decay rates generic differences appear not only between the power-law exponents for the midway rank decline but also for small and large rank. We extend the theoretical framework to include thermodynamic and statistical-mechanical concepts, such as entropies and configuration.

A mere hyperbolic law, like the Zipf's law power function, is often inadequate to describe rank-size relationships. An alternative theoretical distribution is proposed based on theoretical physics arguments starting from the Yule-Simon distribution. A modeling is proposed leading to a universal form. A theoretical suggestion for the "best (or optimal) distribution", is provided through an entropy argument. The ranking of areas through the number of cities in various countries and some sport competition ranking serves for the present illustrations., Comment: 17 pages, 7 figures, 2 Tables, 49 references

In spite of decades of theorizing, the origins of Zipf's law remain elusive. I propose that a Zipfian distribution straightforwardly follows from the interaction of syntax (word classes differing in class size) and semantics (words having to be sufficiently specific to be distinctive and sufficiently general to be reusable). These factors are independently motivated and well-established ingredients of a natural-language system. Using a computational model, it is shown that neither of these ingredients suffices to produce a Zipfian distribution on its own and that the results deviate from the Zipfian ideal only in the same way as natural language itself does.

A series of recent works studying a database of chronologically sorted chess games --containing 1.4 million games played by humans between 1998 and 2007-- have shown that the popularity distribution of chess game-lines follows a Zipf's law, and that time series inferred from the sequences of those game-lines exhibit long-range memory effects. The presence of Zipf's law together with long-range memory effects was observed in several systems, however, the simultaneous emergence of these two phenomena were always studied separately up to now. In this work, by making use of a variant of the Yule-Simon preferential growth model, introduced by Cattuto et al., we provide an explanation for the simultaneous emergence of Zipf's law and long-range correlations memory effects in a chess database. We find that Cattuto's Model (CM) is able to reproduce both, Zipf's law and the long-range correlations, including size-dependent scaling of the Hurst exponent for the corresponding time series. CM allows an explanation for the simultaneous emergence of these two phenomena via a preferential growth dynamics, including a memory kernel, in the popularity distribution of chess game-lines. This mechanism results in an aging process in the chess game-line choice as the database grows. Moreover, we find burstiness in the activity of subsets of the most active players, although the aggregated activity of the pool of players displays inter-event times without burstiness. We show that CM is not able to produce time series with bursty behavior providing evidence that burstiness is not required for the explanation of the long-range correlation effects in the chess database., Comment: 18 pages, 7 figures

We study rank-frequency relations for phonemes, the minimal units that still relate to linguistic meaning. We show that these relations can be described by the Dirichlet distribution, a direct analogue of the ideal-gas model in statistical mechanics. This description allows us to demonstrate that the rank-frequency relations for phonemes of a text do depend on its author. The author-dependency effect is not caused by the author's vocabulary (common words used in different texts), and is confirmed by several alternative means. This suggests that it can be directly related to phonemes. These features contrast to rank-frequency relations for words, which are both author and text independent and are governed by the Zipf's law., 16 pages, 4 figures

The word-frequency distribution of a text written by an author is well accounted for by a maximum entropy distribution, the RGF (random group formation)-prediction. The RGF-distribution is completely determined by the a priori values of the total number of words in the text (M), the number of distinct words (N) and the number of repetitions of the most common word (k_max). It is here shown that this maximum entropy prediction also describes a text written in Chinese characters. In particular it is shown that although the same Chinese text written in words and Chinese characters have quite differently shaped distributions, they are nevertheless both well predicted by their respective three a priori characteristic values. It is pointed out that this is analogous to the change in the shape of the distribution when translating a given text to another language. Another consequence of the RGF-prediction is that taking a part of a long text will change the input parameters (M, N, k_max) and consequently also the shape of the frequency distribution. This is explicitly confirmed for texts written in Chinese characters. Since the RGF-prediction has no system-specific information beyond the three a priori values (M, N, k_max), any specific language characteristic has to be sought in systematic deviations from the RGF-prediction and the measured frequencies. One such systematic deviation is identified and, through a statistical information theoretical argument and an extended RGF-model, it is proposed that this deviation is caused by multiple meanings of Chinese characters. The effect is stronger for Chinese characters than for Chinese words. The relation between Zipf's law, the Simon-model for texts and the present results are discussed., Comment: 15 pages, 10 figures, 2 tables

We propose a model that explains the reliable emergence of power laws (e.g., Zipf’s law) during the development of different human languages. The model incorporates the principle of least effort in communications, minimizing a combination of the information-theoretic communication inefficiency and direct signal cost. We prove a general relationship, for all optimal languages, between the signal cost distribution and the resulting distribution of signals. Zipf’s law then emerges for logarithmic signal cost distributions, which is the cost distribution expected for words constructed from letters or phonemes.

A fundamental problem in linguistics is how literary texts can be quantified mathematically. It is well known that the frequency of a (rare) word in a text is roughly inverse proportional to its rank (Zipf's law). Here we address the complementary question, if also the rhythm of the text, characterized by the arrangement of the rare words in the text, can be quantified mathematically in a similar basic way. To this end, we consider representative classic single-authored texts from England/Ireland, France, Germany, China, and Japan. In each text, we classify each word by its rank. We focus on the rare words with ranks above some threshold Q and study the lengths of the (return) intervals between them. We find that for all texts considered, the probability SQ(r) that the length of an interval exceeds r, follows a perfect Weibull-function, SQ(r) = exp(-b(β)rβ), with β around 0.7. The return intervals themselves are arranged in a long-range correlated self-similar fashion, where the autocorrelation function CQ(s) of the intervals follows a power law, CQ(s) ∼ s-γ, with an exponent γ between 0.14 and 0.48. We show that these features lead to a pronounced clustering of the rare words in the text.

Urban scaling relations characterizing how diverse properties of cities vary on average with their population size have recently been shown to be a general quantitative property of many urban systems around the world. However, in previous studies the statistics of urban indicators were not analyzed in detail, raising important questions about the full characterization of urban properties and how scaling relations may emerge in these larger contexts. Here, we build a self-consistent statistical framework that characterizes the joint probability distributions of urban indicators and city population sizes across an urban system. To develop this framework empirically we use one of the most granular and stochastic urban indicators available, specifically measuring homicides in cities of Brazil, Colombia and Mexico, three nations with high and fast changing rates of violent crime. We use these data to derive the conditional probability of the number of homicides per year given the population size of a city. To do this we use Bayes' rule together with the estimated conditional probability of city size given their number of homicides and the distribution of total homicides. We then show that scaling laws emerge as expectation values of these conditional statistics. Knowledge of these distributions implies, in turn, a relationship between scaling and population size distribution exponents that can be used to predict Zipf's exponent from urban indicator statistics. Our results also suggest how a general statistical theory of urban indicators may be constructed from the stochastic dynamics of social interaction processes in cities.

Timbre is a key perceptual feature that allows discrimination between different sounds. Timbral sensations are highly dependent on the temporal evolution of the power spectrum of an audio signal. In order to quantitatively characterize such sensations, the shape of the power spectrum has to be encoded in a way that preserves certain physical and perceptual properties. Therefore, it is common practice to encode short-time power spectra using psychoacoustical frequency scales. In this paper, we study and characterize the statistical properties of such encodings, here called timbral code-words. In particular, we report on rank-frequency distributions of timbral code-words extracted from 740 hours of audio coming from disparate sources such as speech, music, and environmental sounds. Analogously to text corpora, we find a heavy-tailed Zipfian distribution with exponent close to one. Importantly, this distribution is found independently of different encoding decisions and regardless of the audio source. Further analysis on the intrinsic characteristics of most and least frequent code-words reveals that the most frequent code-words tend to have a more homogeneous structure. We also find that speech and music databases have specific, distinctive code-words while, in the case of the environmental sounds, this database-specific code-words are not present. Finally, we find that a Yule-Simon process with memory provides a reasonable quantitative approximation for our data, suggesting the existence of a common simple generative mechanism for all considered sound sources. © 2012 Haro et al., Funding was received from Classical Planet: TSI-070100- 2009-407 (MITYC), www.mityc.es; DRIMS: TIN2009-14247-C02-01 (MICINN), www.micinn.es; FIS2009-09508, www.micinn.es; and 2009SGR-164, www.gencat.cat. JS acknowledges funding from Consejo Superior de Investigaciones Cientı´ficas (JAEDOC069/2010), www.csic.es; and Generalitat de Catalunya (2009-SGR-1434), www.gencat.cat.

Background Zipf's discovery that word frequency distributions obey a power law established parallels between biological and physical processes, and language, laying the groundwork for a complex systems perspective on human communication. More recent research has also identified scaling regularities in the dynamics underlying the successive occurrences of events, suggesting the possibility of similar findings for language as well. Methodology/Principal Findings By considering frequent words in USENET discussion groups and in disparate databases where the language has different levels of formality, here we show that the distributions of distances between successive occurrences of the same word display bursty deviations from a Poisson process and are well characterized by a stretched exponential (Weibull) scaling. The extent of this deviation depends strongly on semantic type – a measure of the logicality of each word – and less strongly on frequency. We develop a generative model of this behavior that fully determines the dynamics of word usage. Conclusions/Significance Recurrence patterns of words are well described by a stretched exponential distribution of recurrence times, an empirical scaling that cannot be anticipated from Zipf's law. Because the use of words provides a uniquely precise and powerful lens on human thought and activity, our findings also have implications for other overt manifestations of collective human dynamics.

Urban areas and their voracious appetites are increasingly dominating the flows of energy and materials around the globe. Understanding the size distribution and dynamics of urban areas is vital if we are to manage their growth and mitigate their negative impacts on global ecosystems. For over 50 years, city size distributions have been assumed to universally follow a power function, and many theories have been put forth to explain what has become known as Zipf's law (the instance where the exponent of the power function equals unity). Most previous studies, however, only include the largest cities that comprise the tail of the distribution. Here we show that national, regional and continental city size distributions, whether based on census data or inferred from cluster areas of remotely-sensed nighttime lights, are in fact lognormally distributed through the majority of cities and only approach power functions for the largest cities in the distribution tails. To explore generating processes, we use a simple model incorporating only two basic human dynamics, migration and reproduction, that nonetheless generates distributions very similar to those found empirically. Our results suggest that macroscopic patterns of human settlements may be far more constrained by fundamental ecological principles than more fine-scale socioeconomic factors.

In linguistic studies, the academic level of the vocabulary in a text can be described in terms of statistical physics by using a ''temperature'' concept related to the text's word-frequency distribution. We propose a ''comparative thermo-linguistic'' technique to analyze the vocabulary of a text to determine its academic level and its target readership in any given language. We apply this technique to a large number of books by several authors and examine how the vocabulary of a text changes when it is translated from one language to another. Unlike the uniform results produced using the Zipf law, using our ''word energy'' distribution technique we find variations in the power-law behavior. We also examine some common features that span across languages and identify some intriguing questions concerning how to determine when a text is suitable for its intended readership. Copyright

Zipf's law is one the most conspicuous empirical facts for cities, however, there is no convincing explanation for the scaling relation between rank and size and its scaling exponent. Based on the idea from general fractals and scaling, this paper proposes a dual competition hypothesis of city develop to explain the value intervals and the special value, 1, of the power exponent. Zipf's law and Pareto's law can be mathematically transformed into one another. Based on the Pareto distribution, a frequency correlation function can be constructed. By scaling analysis and multifractals spectrum, the parameter interval of Pareto exponent is derived as (0.5, 1]; Based on the Zipf distribution, a size correlation function can be built, and it is opposite to the first one. By the second correlation function and multifractals notion, the Pareto exponent interval is derived as [1, 2). Thus the process of urban evolution falls into two effects: one is Pareto effect indicating city number increase (external complexity), and the other Zipf effect indicating city size growth (internal complexity). Because of struggle of the two effects, the scaling exponent varies from 0.5 to 2; but if the two effects reach equilibrium with each other, the scaling exponent approaches 1. A series of mathematical experiments on hierarchical correlation are employed to verify the models and a conclusion can be drawn that if cities in a given region follow Zipf's law, the frequency and size correlations will follow the scaling law. This theory can be generalized to interpret the inverse power-law distributions in various fields of physical and social sciences., 18 pages, 3 figures, 3 tables

Written text is one of the fundamental manifestations of human language, and the study of its universal regularities can give clues about how our brains process information and how we, as a society, organize and share it. Among these regularities, only Zipf's law has been explored in depth. Other basic properties, such as the existence of bursts of rare words in specific documents, have only been studied independently of each other and mainly by descriptive models. As a consequence, there is a lack of understanding of linguistic processes as complex emergent phenomena. Beyond Zipf's law for word frequencies, here we focus on burstiness, Heaps' law describing the sublinear growth of vocabulary size with the length of a document, and the topicality of document collections, which encode correlations within and across documents absent in random null models. We introduce and validate a generative model that explains the simultaneous emergence of all these patterns from simple rules. As a result, we find a connection between the bursty nature of rare words and the topical organization of texts and identify dynamic word ranking and memory across documents as key mechanisms explaining the non trivial organization of written text. Our research can have broad implications and practical applications in computer science, cognitive science and linguistics.

The results from urban scaling in recent years have held the promise of increased efficiency to the societies who could actively control the distribution of their cities' size. However, little evidence exists as to the factors which influence the level of urban unevenness, as expressed by the slope of the rank-size distribution, partly because the diversity of results found in the literature follows the heterogeneity of analysis specifications. In this study, I set up a meta-analysis of Zipf's law which accounts for technical as well as topical factors of variations of Zipf's coefficient. I found 86 studies publishing at least one empirical estimation of this coefficient and recorded their metadata into an open database. I regressed the 1962 corresponding estimates with variables describing the study and the estimation process as well as socio-demographic variables describing the territory under enquiry. A dynamic meta-analysis was also performed to look for factors of evolution of city size unevenness. The results of the most interesting models are presented in the article, whereas all analyses can be reproduced on a dedicated online platform. The results show that on average, 40% of the variation of Zipf's coefficients is due to the technical choices. The main other variables associated with distinct evolutions are linked to the urbanisation process rather than the process of economic development and population growth. Finally, no evidence was found to support the effectiveness of past planning actions in modifying this urban feature.