Search

• The high frequencies of CFTR variants can be explained by a selective advantage for healthy carriers (4–6 % per generation). • Spreading of disease-causing allele carriers was accelerated towards the end of the 18th century. • The most widespread variant (F508del) is estimated to have appeared approximately 190 generations ago. The French region of Brittany presents one of the highest cystic fibrosis (CF)-causing allele frequency in Europe. Here, we tested two hypotheses: i) CF-causing allele carriers arrived by sea in the middle of the 1st millennium AD, and ii) a selective advantage for healthy carriers explains this high rate. From the census of cystic fibrosis patients, frequency maps of the most widespread alleles were established. A mathematical model was developed, based on birth date and place of the ancestors of these patients over 5 centuries, to determine the distribution of local migrations and their parameters for inter-generational intervals of 32 years. This model was applied to simulate the spread of CF-causing variant carriers, according to different scenarios that corresponded to the immigration of a given number of variant carriers at different times (year) and places, and to compare their results to current frequency maps. Migrants carrying a CFTR variant settled in several locations, around which they spread, notably in Central Brittany (F508del variant), Léon (G551D variant) and Cornouaille (1078delT variant). Until the end of the 18th century, the spreading of disease-causing allele carriers was relatively slow, and allele frequencies progressively increased. Then, the mean migration distances increased rapidly, leading to a decline in local frequencies. The main CFTR variants could only have reached their current frequencies through a selective advantage for healthy carriers of the order of 4–6 % at each generation. For the most widespread variant (F508del), the model supports the hypothesis that it appeared around 190 generations ago. [ABSTRACT FROM AUTHOR]

We investigate the market value of rarity theoretically and empirically. Prior studies find that the market value of rarity follows a power law, but this finding lacks a theoretical foundation. We provide a micro-foundation for this finding, demonstrating that the observed power law emerges in a competitive market where agents have rank-dependent utility preferences. The model leads to two new theoretical insights: (i) the rank of an item within a set of close substitutes and the quantity of that item known to exist are both natural measures of rarity, but rank is predicted to perform better; (ii) there is a systematic relationship between the estimated slope and intercept from a regression of log price on log rank. When we test the model on data from over 4000 auction records of rare baseball cards, we find that a regression with only log rank and grading company explains 60% of the variation in log price. [ABSTRACT FROM AUTHOR]

The science of complexity aims to answer the question of what rules nature chooses when assembling the basic constituents of matter and energy into structures and dynamical patterns that cascade through the entire hierarchy of scales in the Universe. A related phenomenon – natural language – can successfully mirror such structures as reflected by its ability to encode and transmit information about them and among them. It is thus legitimate to expect that natural language carries the essence of complexity. And indeed, in the human's speaking and writing it is particularly true that more is different. Natural language thus deserves a central place in the related quantitative study within the science of complexity. With this in mind the present review summarizes the main methodological concepts used in this domain and documents their applicability and utility in identifying universal as well as system-specific features of natural language in its written representation in several major Western languages. In particular, three main complexity-related current research trends in quantitative linguistics are exhaustively covered. The first part addresses the issue of word frequencies in texts and, in particular, demonstrates that taking punctuation into consideration largely restores scaling whose violation in the Zipf's law for the most frequent words is commonly modelled by the so-called Mandelbrot's correction. The second part introduces methods inspired by time series analysis, used in studying various kinds of long-range correlations in written texts. The related time series are generated on the basis of text partition into sentences or into phrases between consecutive punctuation marks. It turns out that these series develop features often found in signals generated by complex systems: the presence of long-range correlations along with fractal or even multifractal structures. Moreover, it appears that the distances between consecutive punctuation marks quite universally across languages comply with the discrete variant of the Weibull distribution, often appearing in survival analysis. In the third part, the application of the network formalism to natural language is reviewed, particularly in the context of word-adjacency networks whose structure reflects the word co-occurrence in texts. Various parameters characterizing topology of such networks can be used for classification of texts, for example, from a stylometric perspective. Network approach can also be applied in semantic analysis to represent a hierarchy of words and associations between them based on their meaning. Structure of such networks turns out to be significantly different from that observed in random networks, revealing genuine properties of language. Finally, punctuation appears to have a significant impact not only on the language's information-carrying ability but also on its key statistical properties, hence it seems recommended to consider punctuation marks on a par with words. [ABSTRACT FROM AUTHOR]

Dependency distance (DD) traditionally refers to the linear distance between two related words in the dependency structure of a sentence (i.e., syntactic-DD). Based on this concept, previous studies have found that language follows a universal rule of DD minimization (DDM), which states that short DDs occur more frequently than long DDs. But if the DDM as a structural rule is universal, does it extend to other levels of language structure? If so, are there any individualities at different levels? This paper proposes the concept of discourse-DD, i.e., the linear distance between two related clauses in a text, to investigate whether the DDM exists at the discourse level. We present a statistical study of discourse-DD using a discourse dependency corpus and compare the statistical results with the syntactic structure. The results show that discourse structure does indeed show the DDM tendency, and even more strongly than syntactic structure. The reasons for this may be that discourse patterns are more flexible than syntactic patterns and clause functions are more dynamic than word functions, and in this case, a stronger DDM is required for discourse comprehension. This paper also explores how linguistic factors such as discourse length, complex sentences, and connectives influence discourse-DDM. The paper deepens the research on DDM of language. • Extending dependency distance minimization study from syntax to discourse to develop a more general language rule. • Proposing the concept of discourse dependency distance and investigate it statistically using a discourse dependency corpus. • Claiming a more stronger dependency distance minimization tendency of discourse than syntax based on statistical comparison. • Giving reasons for stronger dependency distance minimization of discourse from grammatical and cognitive aspects. • Investigating how linguistic factors influence discourse dependency distance minimization. [ABSTRACT FROM AUTHOR]

Following the ideas of prospect theory, a class of growth functions is used to characterize deterministic variations of firm size, which portrays the asymmetric efforts of the firm to achieve the desired size. Considering the differences in the ability of different firm size to cope with uncertainties, the Boltzmann equation for the evolution of firm size is constructed. Utilizing a suitable scaling limit, the Fokker–Planck equation is acquired and its explicit steady-state solution is derived. Our results illustrate that different choices of parameters in the growth function lead to various statistical laws for firm size, such as the Amoroso distribution, the lognormal distribution and Zipf's law. Under certain conditions, inequality for the distribution of firm size decreases as firm size increases. The numerical analyses are presented to illustrate our results. • A class of models for firm size evolution is constructed by using the kinetic theory. • Growth functions with prospect theory ideas are used to depict changes in firm size. • Different choices of parameters in growth function cause different statistical laws. • Inequality of firm size distribution reduces with firm size under certain conditions. [ABSTRACT FROM AUTHOR]

Traditional Chinese medicine (TCM) prescriptions have been used to cure diseases in China for thousands of years, in which many TCM herbs have no definite common quantity. Some key TCM herbs are commonly used and thus deserve in-depth investigations based on a more acceptable classification method. This study analyzes whether TCM prescriptions follow Zipf's law and attempts to obtain the thresholds of key TCM herbs based on the application of Zipf's law. A total of 84,418 TCM prescriptions were collected and standardized. We tested whether Zipf's law and Zipf's distribution fit the Chinese herb distributions. A linear fitting experiment was performed to verify the relationship between the frequency distribution and frequency of TCM herbs. The distribution of TCM herbs in TCM prescriptions conformed to Zipf's law. Accordingly, the thresholds were obtained for the key TCM herbs. The distribution of TCM herbs in TCM prescriptions follows Zipf's law. [ABSTRACT FROM AUTHOR]

Neighbourhood roads are critical for developing walkable cities and improving residents' health. This study introduces a prototype system, the Road Evaluation by Desire Path Simulation System (RED-PaSS), based on agent-based modelling (ABM) designed to assess the alignment of roads with pedestrian natural movement. RED-PaSS simulates pedestrian movement to generate optimal desire path systems, representing the most direct and natural paths pedestrians would choose in urban settings. The system evaluates existing roads against these simulated desire paths using key metrics, including road patch distance, road segment alignment with desire paths, and origin–destination (OD) flow efficiency and compares road networks using a ranking index. In the case study, the RED-PaSS was applied to evaluate and rank 708 US neighbourhood-scale road networks using an open-source road dataset from Zillow. Results showed that the roads closely aligned with the simulated desire paths offer more direct and efficient pedestrian routes, enhancing their natural walking movement. In contrast, poorly aligned roads often force pedestrians to take indirect routes, reducing their walkability and efficiency. Incorporating optimal desire paths with road networks also reduces average pedestrian walking distance (for example, by approximately 28%, decreasing from 43 to 31 metres in a sample neighbourhood). Moreover, the ranking of the 708 road networks demonstrated Zipf's law distribution R 2 = 0.95 , indicating that the majority have poor alignment with their optimal desire paths while very few align well. The top-ranked neighbourhoods demonstrated higher alignment scores and better coverage, effectively connecting areas with high pedestrian demand. The RED-PaSS offers actionable insights for designing walkable cities by predicting optimal desire paths and identifying discrepancies between existing roads and the optimal, which can serve as a valuable tool for urban planners creating more pedestrian-friendly urban environments. [ABSTRACT FROM AUTHOR]

Beyond the conventional focus on regional industrial decline and deterioration, the increasing prevalence of shrinking cities worldwide calls for a structural reevaluation. This study employs Zipf's law to establish a national-level quantitative empirical framework for analyzing the evolving dynamics of shrinking cities, incorporating various stages of urban life cycles. Japan offers an ideal context for analysis, showcasing a complete urban evolution within a closed system. Our findings indicate that, within the dynamics of city size distribution, the ongoing growth of large cities is accompanied by the increasing incidence of shrinking cities. We introduce the "Urban Growth Divides" hypothesis to explain the widening disparities between large and small cities. A consistent shift in the Zipf exponent shows the intensification of the divides, further amplified by demographic decline. The strong correlation between population change rates and the Zipf exponent confirms the inevitability of deepening systemic differentiation. This research underscores the necessity of analyzing shrinking cities through the lens of systemic urbanization evolution, particularly highlighting the inevitability of Urban Growth Divides driven by demographic dynamics. It advocates for strategic policy responses to the structural challenges of shrinking cities in an era of population mobility. • It adopts a transtemporal perspective on shrinking cities, emphasizing the need for a systemic reevaluation amid their growing prevalence. • It constructs a national-level, quantifiable empirical analysis framework to capture the dynamics of shrinking cities. • It suggests that shrinking cities, in the deepening "Urban Growth Divides" trend, are structural, not just industrial. • The findings indicate a significant correlation between population decline and increasing city size disparities. • The study advocates for strategic policy responses to the structural challenges of shrinking cities in an era of population mobility. [ABSTRACT FROM AUTHOR]

Rank-size distributions, such as Zipf's Law, have been instrumental in providing insights into the emergence of hierarchies across diverse systems, from linguistic corpuses to urban structures. However, the application of Zipf's Law reveals limitations, particularly in its focus on distribution tails, sometimes overlooking a large proportion of the data which might play a pivotal role in system dynamics. Yet, fitting rank-size distributions other than a straight line on the log–log scale requires caution. In this study, we re-evaluate the utility of rank-size distributions by contrasting the traditional Zipf's Law with the Discrete Generalized Beta Distribution (DGBD). We show the need of cautious fitting techniques for rank distributions, including the use of binning to prevent overfitting to data tails. Through both analytical derivation and empirical validation on commit data of open-source repositories, we show that DGBD consistently improves over Zipf distribution for concave rank distributions of large datasets (N ≥ 100). This approach contributes to the advancement of methodologies for analyzing hierarchical systems. [ABSTRACT FROM AUTHOR]

Learning is a distinctive feature of intelligent behaviour. High-throughput experimental data and Big Data promise to open new windows on complex systems such as cells, the brain or our societies. Yet, the puzzling success of Artificial Intelligence and Machine Learning shows that we still have a poor conceptual understanding of learning. These applications push statistical inference into uncharted territories where data is high-dimensional and scarce, and prior information on "true" models is scant if not totally absent. Here we review recent progress on understanding learning, based on the notion of "relevance". The relevance, as we define it here, quantifies the amount of information that a dataset or the internal representation of a learning machine contains on the generative model of the data. This allows us to define maximally informative samples, on one hand, and optimal learning machines on the other. These are ideal limits of samples and of machines, that contain the maximal amount of information about the unknown generative process, at a given resolution (or level of compression). Both ideal limits exhibit critical features in the statistical sense: Maximally informative samples are characterised by a power-law frequency distribution (statistical criticality) and optimal learning machines by an anomalously large susceptibility. The trade-off between resolution (i.e. compression) and relevance distinguishes the regime of noisy representations from that of lossy compression. These are separated by a special point characterised by Zipf's law statistics. This identifies samples obeying Zipf's law as the most compressed loss-less representations that are optimal in the sense of maximal relevance. Criticality in optimal learning machines manifests in an exponential degeneracy of energy levels, that leads to unusual thermodynamic properties. This distinctive feature is consistent with the invariance of the classification under coarse graining of the output, which is a desirable property of learning machines. This theoretical framework is corroborated by empirical analysis showing (i) how the concept of relevance can be useful to identify relevant variables in high-dimensional inference and (ii) that widely used machine learning architectures approach reasonably well the ideal limit of optimal learning machines, within the limits of the data with which they are trained. [ABSTRACT FROM AUTHOR]

Annotated corpus can greatly assist in the natural language processing field. For example, computers can understand more of the document context, and indexing and clustering in information retrieval can be done precisely with less or no ambiguity of words. However, there are only a few annotated corpora in Malay language, which are not publicly shared. In this paper, we delve into analysing and annotating Malay translated hadith documents in terms of tagging and entities. There are three phases, which are manual filtering and cleaning, analysing the corpus and creating the benchmark. As the result, an analysis and benchmark of Malay translated hadith corpus were produced in term of part-of-speech and named entities tags that follows the Zipf's law distribution. [ABSTRACT FROM AUTHOR]

Consider a society comprising n members that are ranked in a decreasing order of their personal wealths. For this society: Zipf's Law manifests the case in which the members' ranks and wealths display an inverse power-law relation; and Price's Law manifests the case in which the n richest members possess, collectively, half of all the wealth. This paper goes from Zipf's Law to Price's Law and beyond, and it does so by introducing and exploring a novel Generalized Price's Law (GPL): a general allometric scaling of the quantiles of rank distributions. Akin to multifractals, the GPL is governed by spectrums of powers. The GPL spectra are investigated, and are shown to be fractal objects: from a socioeconomic perspective, the spectra are power-law Lorenz curves that characterize poor fractality and rich fractality; from a probabilistic perspective, the spectra are characterized by Pareto statistics and by Lindy Laws. The intersection of Zipf's Law and of the GPL is also investigated, and it is shown to be: (i) the phase transition between the two markedly different fractal regimes of the GPL spectra; (ii) the phase transition between two markedly different macroscopic regimes of Zipf's Law; (iii) Price's Law. Metaphorically, this paper establishes navigation directions in the space of rank distributions: how to get from the 'Zipf street' to the 'Price junction', from there to the new 'GPL avenue', and back. [ABSTRACT FROM AUTHOR]

A novel method is proposed to define cities' boundaries within a given national urban system. Point Volunteered Geographic Information, such as the OpenStreetMap road network nodes, together with national geospatial information, like economic units, are recursively clustered using DBSCAN at different distance thresholds; clusters smaller than a threshold reflecting the rural settlements pattern are filtered, and then Shannon's entropy for the entire system is calculated. The system of percolated nodes obtained when this entropy reaches its maximum is compared against the current definition of what is officially urban. It is found to be very similar to the actual urban-nonurban classification, and also to have a very high correspondence with the current classification of the Mexican National Urban System (SUN). An additional finding is that the percolated system at the maximum entropy value has a very high correspondence with Zipf's law, while the current Mexican SUN does not. [Display omitted] • We define urban boundaries with a combination of crowdsourced and official datasets using DBSCAN. • The criteria are rural cluster size threshold and the distance at which Shannon's entropy index maximum was found. • The percolated clusters fit Zipf's Law better than the National Urban System (SUN). • We obtained a global similarity of around 86% and locally above 75% with only five cities missing. [ABSTRACT FROM AUTHOR]

We test the validity of Zipf's and Gibrat's laws for city size distributions at the regional level from 1900 to 2010 by considering US states. Zipf's law is satisfied for a majority of states, but for the United States as a whole it only held during the first half of the twentieth century. The null hypothesis of a power law is not rejected at the national level or for most states (the maximum number of rejections in one year is 13 states out of 48). There is evidence supporting a weak version of Gibrat's law in the long-term; mean growth is independent of initial population for most city sizes over the entire United States and in 27 states, while the variance of growth is size-dependent. • We test the validity of Zipf's and Gibrat's laws for city size distributions at the US regional level from 1900–2010. • Zipf's law holds for a majority of states, but for the US as a whole it only held during the first half of the 20th century. • The null hypothesis of a power law is not rejected at the national level or for most states. • There is evidence supporting a weak version of Gibrat's law in the long-term. • Mean growth is independent of initial population for most city sizes over the entire US and in 27 states. [ABSTRACT FROM AUTHOR]

Exploiting the hierarchical structure of cities and based on a dataset for U.S. cities between 1840 and 2016, the aim of this paper is to analyze the evolution of the U.S. city size distribution. For that purpose we estimate a general three‐parameter Zipf model, which can be traced back to Mandelbrot (1982), and validate our results by means of the hierarchical scaling law. Especially in the second half of the twentieth century, we find a pronounced departure from the exact Zipf's law. The city size distribution has become more equally distributed over time. Besides, the applied estimation method reveals evidence for leading cities dominating the remaining largest cities. Thus, the growing equality of the city sizes can be explained rather by growing smaller cities than by a loss of importance of the largest ones. [ABSTRACT FROM AUTHOR]

• Hyperbolic geometry describes multiple aspects of human perception. • Hyperbolic geometry describes multiple aspects of signals from natural environments. • Correspondence between Zipf's law and hyperbolic geometry. • Maximally informative representations lead to hyperbolic geometry. This review connects several lines of research to argue that hyperbolic geometry should be broadly applicable to neural circuits as well as other biological circuits. The reason for this is that networks that conform to hyperbolic geometry are maximally responsive to external and internal perturbations. These networks also allow for efficient communication under conditions where nodes are added or removed. We will argue that one of the signatures of hyperbolic geometry is the celebrated Zipf's law (also sometimes known as the Pareto distribution) that states that the probability to observe a given pattern is inversely related to its rank. Zipf's law is observed in a variety of biological systems — from protein sequences, neural networks to economics. These observations provide further evidence for the ubiquity of networks with an underlying hyperbolic metric structure. Recent studies in neuroscience specifically point to the relevance of a three-dimensional hyperbolic space for neural signaling. The three-dimensional hyperbolic space may confer additional robustness compared to other dimensions. We illustrate how the use of hyperbolic coordinates revealed a novel topographic organization within the olfactory system. The use of such coordinates may facilitate representation of relevant signals elsewhere in the brain. [ABSTRACT FROM AUTHOR]

• We introduce a hyperparameter β and conduct power transformation to calibrate the GloVe model. • We present an estimation method of the hyperparameter β for the different sizes of corpus. • Our proposed method outperforms the original GloVe model on word analogy and word similarity tasks. • Our method has a higher convergence speed than GloVe. GloVe is a popular method for learning word representation by fitting log-bilinear model. However, many researchers have observed that the number of co-occurrences of word pairs in Glove generally follow power law distribution with the value of power law exponent larger than 1; and the value becomes increasingly larger with the increase of corpus size, which is inconsistent with the principle of Zipf's law. Zipf's law is a power law distribution with the value of power law exponent equal to 1 and is a basic law of corpus linguistics. On the basis of this principle, this study aims to verify whether the performance of GloVe can be improved after processing certain calibrations on the GloVe model. The estimated value of power law exponent is generally obtained via least square estimation in the linear regression for the logarithm of the number of co-occurrences with the logarithm of their ranks. Inspired by processing a suitable power transformation of response variable in linear regression model to improve the model fitting, we implement power transformation to the logarithm of the number of co-occurrences of word pairs [i.e. (logX ij) β ] and appropriately select β , such that the estimated value of power law exponent approximates to 1. An estimation method of the hyperparameter β is presented for the different sizes of corpus. Then, we use (logX ij) β instead of logX ij in GloVe to learn the distributed representations of words. We evaluate this new approach on word analogy and word similarity tasks. Experimental results show that our method outperforms GloVe on the word analogy task, and the improvement of performance is significant under McNemar's test. For word similarity task, the proposed method also outperforms GloVe on five benchmark datasets. Furthermore, our method has a higher convergence speed than GloVe. [ABSTRACT FROM AUTHOR]

We present a kinetic approach to the formation of urban agglomerations which is based on simple rules of immigration and emigration. In most cases, the Boltzmann-type kinetic description allows to obtain, within an asymptotic procedure, a Fokker–Planck equation with variable coefficients of diffusion and drift, which describes the evolution in time of some probability density of the city size. It is shown that, in dependence of the microscopic rules of migration, the equilibrium density can follow both a power law for large values of the size variable, which contains as particular case a Zipf's law behavior, and a lognormal law for middle and low values of the size variable. In particular, connections between the value of Pareto index of the power law at equilibrium and the disposal of the population to emigration are outlined. The theoretical findings are tested with recent data of the populations of Italy and Switzerland. • We introduce a linear kinetic model for the size distribution of cities. • The elementary interactions describe the rates of immigration. • The limit of small interactions results in a Fokker–Planck equation. • The steady state can be both a tailed density and a lognormal one. • The lognormal shows a good fitting with real data from Italy and Swiss. [ABSTRACT FROM AUTHOR]

The paper explores the frequency and size distributions of firm-size in a novel dataset for the mid-Victorian era from a recent extraction of the England and Wales population censuses of 1851, 1861, 1871, and 1881. The paper contrasts the hypothesis of the Power Laws against the Lognormal model for the tails of the distributions using maximum likelihood estimation, log likelihood ratio, clipped sample coefficient of variation UMPU-Wilks test, Kolmogorov–Smirnov statistic, among other state-of-the-art statistical methods. Our results show that the Power Law hypothesis is accepted for the size distribution for the years 1851 and 1861, while 1871 is marginally non-significant, but for 1881 the test is inconclusive. The paper discusses the process that generates these distributions citing recent literature that shows how after adding an i.i.d. noise to the Gibrat's multiplicative model one can recreate a Power Law behaviour. Overall, the paper provides, describes and statistically tests for the very first time a unique historical dataset confirming that the tails of the distributions at least for 1851 and 1861 follow a Pareto model and that the Lognormal model is firmly rejected. • We test the tail behaviour of mid-Victorian firm-size distributions. • Pareto model fits the tail of the size distribution of firm size for 1851 and 1861. • The tests are inconclusive for the frequency distributions. • Modern data show a fattening of the tails. [ABSTRACT FROM AUTHOR]

This paper analyses the evolution of the European urban system from a long‐term perspective (from 1300 to 1800). Using the method recently proposed by Clauset, Shalizi, and Newman, a Pareto‐type city size distribution (power law) is rejected from 1300 to 1600. A power law is a plausible model for the city size distribution only in 1700 and 1800, although the log‐normal distribution is another plausible alternative model that we cannot reject. Moreover, the random growth of cities is rejected using parametric and non‐parametric methods. The results reveal a clear pattern of convergent growth in all the periods. [ABSTRACT FROM AUTHOR]

From an informetrics perspective, we discuss the relationship between the Least Effort Principle and information cost minimisation, and discuss the implications for Zipf's law. We propose a mathematical proof of a theorem demonstrating that if one minimises the average cost per unit of information, the Least Effort Principle (LEP) will be verified regardless of the effort function chosen. We then show that Zipf's law can be demonstrated on the basis of the two models studied, namely Average Cost per Unit of Information and the Least Effort Principle (LEP). • Math proof shows minimizing info cost per unit satisfies Least Effort Principle (LEP) for any effort function chosen. • In the continuous mode, the equivalence of the two principles is shown using a class of 'admissible' effort functions. • This article establishes the relationship between the PLE and the average cost per unit of information. • If we minimise the average cost per unit of information, PLE is proved independently of the chosen effort function. [ABSTRACT FROM AUTHOR]

Abstract This paper examines the Zipf's law for the size distribution of Pakistani cities using the five census data from 1951 to 1998. We observe that Zipf's law does not hold in any of the five census years at national level. Next, we consider the city size distribution of four Pakistani provinces and find that Zipf's law is more likely to hold for cities at province-level. We attribute these findings to the coherence property of the urban system. In Pakistan the urban systems within provinces are more coherent in terms of common language, common culture and common rules as compared to the urban system as a whole at national-level. Highlights • Zipf's law is examined at national and four provinces-level for Pakistani cities. • City size distribution does not follow Zipf's law at national-level. • City size distribution is more likely to follow Zipf's law at four provinces-level. • Findings are explained based on the coherence property of the urban system. • Urban systems are more coherent in terms of common language, culture and rules at province-level. [ABSTRACT FROM AUTHOR]

Abstract Luckstead and Devadoss (2017) fitted a distribution made up of lower tail Pareto, lognormal body, and upper tail Pareto to city sizes in the US. Here, we propose a class of distributions providing better fits. The fits are assessed in terms of Akaike Information Criterion, Bayesian Information Criterion and probability plots. Highlights • A class of distributions proposed for modelling US city sizes. • The proposed class provides better fits compared to Luckstead and Devadoss (2017). • The quality of fits assessed in terms of AIC, BIC and probability plots. [ABSTRACT FROM AUTHOR]

Abstract “Granularity” refers to the fact that economies are populated by a few large companies (the big “grains”) that coexist with many smaller companies. Such a distribution of firm sizes is modeled by power laws. This study adds to the international evidence of the granularity hypothesis by considering data for the top 1,000 Brazilian companies. We sort the companies from top to bottom in terms of their net revenues. Then, we adjust power laws to the data and estimate Pareto and Gumbel exponents. We find we cannot dismiss the hypothesis of granularity for the Brazilian companies. We also find the Pareto exponent is approximately one (1.070 ±.015), roughly a Zipf’s law. Such a result is in line with the previous one found for American companies where the Pareto exponent = 1.059. We also find a power-law progress curve best fits the data. Highlights • We assess the granularity hypothesis by considering data for Brazilian companies. • We adjust power laws to the data and estimate Pareto and Gumbel exponents. • We find we cannot dismiss the hypothesis of granularity. • We also find the Pareto exponent is approximately one, roughly a Zipf’s law. • We find a power-law progress curve fits the data even better. [ABSTRACT FROM AUTHOR]

Abstract Morphology of etched silicon wafers during anisotropic etching process is investigated utilizing the geometrical exponents of iso-height contour lines. The images obtained from field emission scanning electron microscopy show pyramidal microstructures which disappear in large time scales and porous silicon structures are formed. The morphology of these microstructures is investigated utilizing the data obtained from atomic force microscopy and contour loop analysis. The results show that the anisotropic etching breaks the up-down symmetry in the height fluctuations of etched silicon surfaces. Area of the contour loops increases by enhancement of pyramids sizes. Fractal dimensions of the loops decrease at the beginning of etching process. Distributions of loops and geometrical exponents of iso-height lines in different etching times are determined. The results indicate that in spite of asymmetry in the height fluctuations, the hyperscaling relation and Zipf's law are verified for the multi-affine silicon surfaces during etching process. [ABSTRACT FROM AUTHOR]

The differential pore lengths L i , the differential pore numbers N i and the differential pore anisotropies B i in porous materials are estimated in a unified way and presented as a function of pore radius r i . Those parameters can be determined from the differential specific surface area S i and the differential specific pore volume V i calculated via nitrogen porosimetry, assuming cylindrical pores. The differential pore length is estimated from relation L i  =  S i 2 /V i  =  N i · l i and corresponds to the total length of N i pores with similar local length l i at each pore group of radius r i estimated at partial pressure (P i /P o ). This parameter bears similarities with the differential pore anisotropy B i given by B i  =  S i 3 /V i 2  =  N i · b i where b i is the local pore anisotropy. Parameter B i is suitable for the ranking of pore numbers vs. pore volumes and, for isotropic cavities with bi  =  1, leads to the ranking of pores according to the Zipf's law. Parameter L i is suitable for the ranking of pore lengths as a function of pore radii and reveals some morphometric similarities between the pore networks in solids and the branching of trees described by the so-called allometric relations. In addition these relations may be used for the distinction of pore number evolution via either power law or exponential mechanisms, expressed by corresponding distributions. Such effects, observed previously by volcanologists in the vesicles of volvanic magmas in mm scale, are also observed in the present study for random pores of nm scale in lab made materials. [ABSTRACT FROM AUTHOR]

Numerical simulations of multiplicatively interacting stochastic processes with weighted selections were conducted. A feedback mechanism to control the weight w of selections was proposed. It becomes evident that when w is moderately controlled around 0, such systems spontaneously exhibit the Pareto–Zipf distribution. The simulation results are universal in the sense that microscopic details, such as parameter values and the type of control and weight, are irrelevant. The central ingredient of the Pareto–Zipf law is argued to be the mild control of interactions. [ABSTRACT FROM AUTHOR]

In this paper, we develop a novel car-following model with inter-vehicle communication to explore each vehicle’s movement in a two-lane traffic system when an incident occurs on a lane. The numerical results show that the proposed model can perfectly describe each vehicle’s motion when an incident occurs, i.e., no collision occurs while the classical full velocity difference (FVD) model produces collision on each lane, which shows the proposed model is more reasonable. The above results can help drivers to reasonably adjust their driving behaviors when an incident occurs in a two-lane traffic system. [ABSTRACT FROM AUTHOR]

Mehri and Jamaati (2017) [18] used Zipf's law to model word frequencies in Holy Bible translations for one hundred live languages. We compare the fit of Zipf's law to a number of Pareto type distributions. The latter distributions are shown to provide the best fit, as judged by a number of comparative plots and error measures. The fit of Zipf's law appears generally poor. [ABSTRACT FROM AUTHOR]

More than half of the world's population currently resides in urban areas. In the majority of developing countries slums are a defining part of the urban scape. Their supply with energy, basic infrastructure, among others is one of the main challenges of modern civilizations. To provide an optimal supply, the spatial patterns of slums in cities have to be explored. While most of current literature is focused on inter-urban dynamics, this paper is focused on intra-urban pattern (i.e. the spatial pattern of morphological slums within a city) and its link to the inter-urban ones. Therefore, census and remote sensing data are analyzed to create rank size distributions of morphological slums in different cities of developing countries. The observations were compared to rank size distributions of cities in a respective developing country. It is found that typical inter-urban pattern can be transferred to intra-urban pattern. Surprisingly is the fact that the size of slums is independent from city and global region in the analyzed cities. The slums in Mumbai, Manila, Rio de Janeiro and Cape Town have an average area of 0.016 km 2 with a standard deviation of only 0.004 km 2 . [ABSTRACT FROM AUTHOR]

The Sandstone Greenstone Belt is an exploration-immature, regolith-covered, approximately 1000 sq. km belt, in the Southern Cross Domain of the Yilgarn Craton. In order to estimate potential endowment, historical gold production and deposit resource estimates are required to be quantitatively analysed for calculation of natural and residual gold endowment. The total residual gold endowment within the oxide zone of the Sandstone Greenstone Belt is estimated, by application of a Zipf’s law statistical assessment, to be 2.3 Moz. This mineralisation is most likely contained in extensions of known deposits and several undiscovered deposits. The fresh rock of the Sandstone Greenstone Belt remains poorly explored. However, a conceptual endowment estimate can be made, based on a minerals system comparison between the exploration-immature Sandstone Greenstone Belt and the well-explored, geologically-similar Agnew Greenstone Belt, 100 km to the east. It is possible that natural endowment at Sandstone could total 21.3 Moz, with nine undiscovered deposits of >0.5 Moz. Application of such a minerals-system integrated endowment assessment represents an effective motivator to embark on a well-resourced gold exploration campaign in the Sandstone Greenstone Belt, a currently immature exploration search space. [ABSTRACT FROM AUTHOR]

Living systems are hierarchical control systems that display a small world network structure. In such structures, many smaller clusters are nested within fewer larger ones, producing a fractal-like structure with a 'power-law' cluster size distribution (a mereology). Just like their structure, the dynamics of living systems shows fractal-like qualities: the timeseries of inner message passing and overt behavior contain high frequencies or 'states' (treble) that are nested within lower frequencies or 'traits' (bass), producing a power-law frequency spectrum that is known as a 'state-trait continuum' in the behavioral sciences. Here, we argue that the power-law dynamics of living systems results from their power-law network structure: organisms 'vertically encode' the deep spatiotemporal structure of their (anticipated) environments, to the effect that many small clusters near the base of the hierarchy produce high frequency signal changes and fewer larger clusters at its top produce ultra-low frequencies. Such ultra-low frequencies exert a tonic regulatory pressure that produces morphological as well as behavioral traits (i.e., body plans and personalities). Nested-modular structure causes higher frequencies to be embedded within lower frequencies, producing a power-law state-trait continuum. At the heart of such dynamics lies the need for efficient energy dissipation through networks of coupled oscillators, which also governs the dynamics of non-living systems (e.q., earthquakes, stock market fluctuations). Since hierarchical structure produces hierarchical dynamics, the development and collapse of hierarchical structure (e.g., during maturation and disease) should leave specific traces in system dynamics (shifts in lower frequencies, i.e. morphological and behavioral traits) that may serve as early warning signs to system failure. The applications of this idea range from (bio)physics and phylogenesis to ontogenesis and clinical medicine. • Living systems universally display hierarchical dynamics (power-law frequency spectra). • We argue that this is due to the hierarchical structure of living systems. • Power-law frequency spectra define state-trait continua (e.g. body plans and personalities). • A collapse of hierarchical control may therefore cause (specific) changes in power-law frequency spectra. • This can serve as an early warning sign for system failure (e.g. physical and mental disorders). [ABSTRACT FROM AUTHOR]

• Zipf's law is exploited for text generation, effectively addressing the challenge of data imbalance in entity extraction. • A novel approach Ziteg is proposed and its effectiveness is demonstrated through extensive experiments in a case of extracting entities from technical reports. • Our discussion highlights the profound impact of informetrics on AI research. We assert that informetrics plays a constructive role in shaping the trajectory and advancement of AI. Entity extraction is critical in the intelligent advancement across diverse domains. Nevertheless, a challenge to its effectiveness arises from the data imbalance, where certain entities are common while others are scarce. To address this issue, this study proposes a novel text generation approach that harnesses Zipf's law, which is a powerful tool from informetrics for studying human language. By employing characteristics of Zipf's law, words within the documents are classified as common and rare ones. Subsequently, sentences are classified into common and rare ones, and are further processed by text generation models accordingly. Rare entities within the generated sentences are then labeled using human-designed rules, serving as a supplement to the raw dataset, thereby mitigating the imbalance problem. The study presents a case of extracting entities from technical documents, and the extensive experimental results on two datasets prove the effectiveness of the proposed method. Furthermore, the significance and potential of Zipf's law in driving the progress of artificial intelligence (AI) is discussed, broadening the scope and coverage of informetrics. By incorporating the foundational principles of informetrics into text generation, this study showcases the pivotal role of informetrics in shaping the design and developmental of AI systems. [ABSTRACT FROM AUTHOR]

Zipf's Law of Abbreviation – the idea that more frequent symbols in a code are simpler than less frequent ones – has been shown to hold at the level of words in many languages. We tested whether it holds at the level of individual written characters. Character complexity is similar to word length in that it requires more cognitive and motor effort for producing and processing more complex symbols. We built a dataset of character complexity and frequency measures covering 27 different writing systems. According to our data, Zipf's Law of Abbreviation holds for every writing system in our dataset — the more frequent characters have lower degrees of complexity and vice-versa. This result provides further evidence of optimization mechanisms shaping communication systems. [ABSTRACT FROM AUTHOR]

This work deals with the construction and testing of metrics of welfare based on behavioral complexity, using assumptions derived from Zipf–Mandelbrot law and graph theory. To test these metrics we compared yellow-breasted capuchins ( Sapajus xanthosternos ) (Wied-Neuwied, 1826) (PRIMATES CEBIDAE) found in two institutions, subjected to different captive conditions: a Zoobotanical Garden (hereafter, ZOO; n = 14 ), in good welfare condition, and a Wildlife Rescue Center (hereafter, WRC; n = 8 ), in poor welfare condition. In the Zipf–Mandelbrot-based analysis, the power law exponent was calculated using behavior frequency values versus behavior rank value. These values allow us to evaluate variations in individual behavioral complexity. For each individual we also constructed a graph using the sequence of behavioral units displayed in each recording (average recording time per individual: 4 h 26 min in the ZOO, 4 h 30 min in the WRC). Then, we calculated the values of the main graph attributes, which allowed us to analyze the complexity of the connectivity of the behaviors displayed in the individuals’ behavioral sequences. We found significant differences between the two groups for the slope values in the Zipf–Mandelbrot analysis. The slope values for the ZOO individuals approached −1, with graphs representing a power law, while the values for the WRC individuals diverged from −1, differing from a power law pattern. Likewise, we found significant differences for the graph attributes average degree, weighted average degree, and clustering coefficient when comparing the ZOO and WRC individual graphs. However, no significant difference was found for the attributes modularity and average path length. Both analyses were effective in detecting differences between the patterns of behavioral complexity in the two groups. The slope values for the ZOO individuals indicated a higher behavioral complexity when compared to the WRC individuals. Similarly, graph construction and the calculation of its attributes values allowed us to show that the complexity of the connectivity among the behaviors was higher in the ZOO than in the WRC individual graphs. These results show that the two measuring approaches introduced and tested in this paper were capable of capturing the differences in welfare levels between the two conditions, as shown by differences in behavioral complexity. [ABSTRACT FROM AUTHOR]

This study provides a systematic review of the existing literature on Zipf’s law for city size distribution. Existing empirical evidence suggests that Zipf’s law is not always observable even for the upper-tail cities of a territory. However, the controversy with empirical findings arises due to sample selection biases, methodological weaknesses and data limitations. The hypothesis of Zipf’s law is more likely to be rejected for the entire city size distribution and, in such case, alternative distributions have been suggested. On the contrary, the hypothesis is more likely to be accepted if better empirical methods are employed and cities are properly defined. The debate is still far from to be conclusive. In addition, we identify four emerging areas in Zipf’s law and city size distribution research including the size distribution of lower-tail cities, the size distribution of cities in sub-national regions, the alternative forms of Zipf’s law, and the relationship between Zipf’s law and the coherence property of the urban system. [ABSTRACT FROM AUTHOR]

While theoretical reasoning emphasizes the complexity in metropolitan spatial structures, empirical efforts still focus on measuring spatial structures using employment centers. This paper measures a metropolitan spatial structure as a system of employment clusters for 361 metro areas in the United States. We apply a log-log model to assess the relationship between a cluster's size and its rank in a metro area. We found that among the largest 50 metro areas: (1) cluster sizes in a metro area follow a power law distribution and (2) larger metro areas tend to have evener spatial structures. The results suggest that policymakers can better predict urban growth locations and sizes; may invest in the largest clusters for the biggest economic payoffs; and should consider holistically all elements' (e.g., clusters, infrastructure, socioeconomic interactions) respective scaling laws in the city for its healthy urban growth. [ABSTRACT FROM AUTHOR]

This paper analyses the relation between socio-economic and institutional factors and the dynamics of city populations, the hierarchy of city systems and the urbanization. Particular attention is devoted on the integration process that several European Countries, often structurally so different, have experimented from the beginning of ’90s. Results show that the hierarchical structures of Member States is more even than expected. Moreover, the analysis have provided evidence that the integration process of the European Union had a mixed impact on the way in which people gathered across the territory of the EU. In details, the entry into force of the Schengen treaty has promoted a process of agglomeration of the population in the larger cities. On the contrary, the introduction of euro has led people to spread over the territory. [ABSTRACT FROM AUTHOR]

Aging means that as things grow old their remaining expected lifetimes lessen. Either faster or slower, most of the things we encounter in our everyday lives age with time. However, there are things that do quite the opposite – they anti-age: as they grow old their remaining expected lifetimes increase rather than decrease. A quantitative formulation of anti-aging is given by the so-called “ Lindy’s Law ”. In this paper we explore Lindy’s Law and its connections to Pareto’s Law, to Zipf’s Law, and to socioeconomic inequality. [ABSTRACT FROM AUTHOR]

The proper characterization of the size distribution and growth of firms represents an important issue in economics and business. We use the Maximum Entropy approach to assess the plausibility of the assumption that firm size follows Lognormal or Pareto distributions, which underlies most recent works on the subject. A comprehensive dataset covering the universe of French firms allows us to draw two major conclusions. First, the Pareto hypothesis for the whole distribution should be rejected. Second, by discriminating across firms based on the number of products sold and markets served, we find that, within the class of multi-product companies active in multiple markets, the distribution converges to a Zipf’s law. Conversely, Lognormal distribution is a good benchmark for small single-product firms. The size distribution of firms largely depends on firms’ diversification patterns. [ABSTRACT FROM AUTHOR]

Zipf's law, as a power-law regularity, confirms long-range correlations between the elements in natural and artificial systems. In this article, this law is evaluated for one hundred live languages. We calculate Zipf's exponent for translations of the holy Bible to several languages, for this purpose. The results show that, the average of Zipf's exponent in studied texts is slightly above unity. All studied languages in some families have Zipf's exponent lower/higher than unity. It seems that geographical distribution impresses the communication between speakers of different languages in a language family, and affect similarity between their Zipf's exponent. The Bible has unique concept regardless of its language, but the discrepancy in grammatical rules and syntactic regularities in applying stop words to make sentences and imply a certain concept, lead to difference in Zipf's exponent for various languages. [ABSTRACT FROM AUTHOR]

City size hierarchy and distribution are always at the heart of urban studies, as they have a special ability to reveal the rules of city development and urban system spatial layouts. There is however a data deficiency with regard to city size hierarchy and distribution; in particular, an absence of complete statistics and spatial differences from the global perspective. To fill this research gap, this paper investigates global city size hierarchy based on 2014 data of more than 190 countries and regions by using classic models of “rank-size” rule, the fractal theory and the law of the primate city. We analyzed the spatial patterns, regional features, and implications for China from multi-scale and multi-dimension perspectives. The results show that: (1) There is an obvious pyramid structure of global city size distribution, but differences exist among countries and regions with different economic development types, suggesting a feature of “various types with pyramid dominated”; (2) The primate feature of global city size distribution is not very obvious. However, the primacy ratios of developed countries are much higher than others, and significant differences exist among different regions; (3) The global Zipf exponent and Hausdorff dimension are 0.66 and 1.29, respectively. Cities with middle ranks are in the majority, and the monopoly power of large and super cities is effective to a certain extent, indicating a decreasing concentration tendency in the city size distribution and a convergence trend in terms of relative population size, especially with regard to the medium and small cities; (4) The medium and small cities develop swiftly with limited agglomeration effect of large cities, and Chinese cities would significantly influence global urban progress and spatial patterns. Therefore, developing 780–800 cities will be reasonable for China's urbanization efforts by 2030. [ABSTRACT FROM AUTHOR]

The linguist George Kingsley Zipf made a now classic observation about the relationship between a word's length and its frequency; the more frequent a word is, the shorter it tends to be. He claimed that this "Law of Abbreviation" is a universal structural property of language. The Law of Abbreviation has since been documented in a wide range of human languages, and extended to animal communication systems and even computer programming languages. Zipf hypothesised that this universal design feature arises as a result of individuals optimising form-meaning mappings under competing pressures to communicate accurately but also efficiently-his famous Principle of Least Effort. In this study, we use a miniature artificial language learning paradigm to provide direct experimental evidence for this explanatory hypothesis. We show that language users optimise form-meaning mappings only when pressures for accuracy and efficiency both operate during a communicative task, supporting Zipf's conjecture that the Principle of Least Effort can explain this universal feature of word length distributions. [ABSTRACT FROM AUTHOR]

In the last decade, given the availability of corpora in several distinct languages, research on multilingual part-of-speech tagging started to grow. Amongst the novelties there is mWANN-Tagger ( m ultilingual w eightless a rtificial n eural n etwork tagger ), a weightless neural part-of-speech tagger capable of being used for mostly-suffix-oriented languages. The tagger was subjected to corpora in eight languages of quite distinct natures and had a remarkable accuracy with very low sample deviation in every one of them, indicating the robustness of weightless neural systems for part-of-speech tagging tasks. However, mWANN-Tagger needed to be tuned for every new corpus, since each one required a different parameter configuration. For mWANN-Tagger to be truly multilingual, it should be usable for any new language with no need of parameter tuning. This article proposes a study that aims to find a relation between the lexical diversity of a language and the parameter configuration that would produce the best performing mWANN-Tagger instance. Preliminary analyses suggested that a single parameter configuration may be applied to the eight aforementioned languages. The mWANN-Tagger instance produced by this configuration was as accurate as the language-dependent ones obtained through tuning. Afterwards, the weightless neural tagger was further subjected to new corpora in languages that range from very isolating to polysynthetic ones. The best performing instances of mWANN-Tagger are again the ones produced by the universal parameter configuration. Hence, mWANN-Tagger can be applied to new corpora with no need of parameter tuning, making it a universal multilingual part-of-speech tagger. Further experiments with Universal Dependencies treebanks reveal that mWANN-Tagger may be extended and that it has potential to outperform most state-of-the-art part-of-speech taggers if better word representations are provided. [ABSTRACT FROM AUTHOR]

This study examines the nexus between high-speed rail (HSR) stations and city evolution from an urban science perspective. We analyzed 1627 HSR stations built from 1976 to 2021 in 533 Chinese and European cities. Node or activity clusters in cities are identified through the street network. Then, the urban forms of Chinese and European cities are compared and the Zipf's law is tested. We found that activity clusters in Europe tend to be more contiguous and clustered, suggesting a clear downtown. In Chinese cities, these clusters are more scattered, suggesting polycentric urban development. When the Zipf's law is applied to the size distribution of activity clusters, the rank-size rule is supported both at the aggregate and country levels. Generally, it takes roughly two decades for an HSR station to be able to mature in attracting related development, such as residence and businesses, to complete the exploration process of city evolution. Finally, contributory factors for the integration of HSR stations with the urban fabrics are examined. Apart from the size of the city, economic power, station features and the spatial pattern of activity clusters, the relative location of the HSR station in relation to the urban form matters. Our study contributes both to the HSR and urban science literature, and has practical implications for locational decisions about HSR stations. • Analyzed 1627 HSR stations built from 1976 to 2021 in 533 cities • Measured activity clusters from an urban science perspective • Urban forms of Chinese and European cities are compared and Zipf's law is tested. • Contributory factors for integration HSR stations with urban fabrics are identified. • City size, GDP, station features, and size of nearest node clusters matter [ABSTRACT FROM AUTHOR]

This paper studies the structure and dynamics of urban agglomeration between 1992 and 2018 in one of most developed regions in China – the Guangdong-Hong Kong-Macao Greater Bay Area (GBA) – based on the harmonized night-time light (NTL) data from the Defense Meteorological Satellite Program (DMSP) and Visible Infrared Imaging Radiometer Suite (VIIRS) imagery. We adopted Zipf's law as the rule to control the spatial clustering of adjacent lit pixels. The clustered pixels as patches are concentration of human activities and can then be used for regional urbanization analysis. Unlike conventional spatial units that are rarely changed, such as administrative boundaries, these patches change subtly each year and, more importantly, can help us to examine precisely the urbanization pattern in both development mode and intensity. We investigated the urban growth pattern during the 27-year timespan and conducted the analysis at both region and city levels. Interestingly, the results suggest that Zipf's law of derived patch sizes can hold nearly every year, showcasing the consistent spatial coherency of the urban agglomeration process in the entire region. The correlation between urbanization metrics for each city at three stages (1992–2000, 2000–2010, and 2010–2018) can further lead to deep insights towards the synergetic course of the coordinated and multi-center development. [Display omitted] • Identify the urbanized area using spatial clustering of time-series NTL data guided by Zipf's law related statistics. • Explore the urban expansion dynamics by conducting a correlation analysis between development mode and intensity. • Confirm the integrated, connected, and polycentric urbanization development process evidenced by the findings. [ABSTRACT FROM AUTHOR]

Image captioning aims to generate natural language descriptions for images. Word occurrences usually obey Zipf's Law, the imbalance phenomenon makes the conventional training bias to majority data. However, this imbalance distribution has not been considered adequately in captioning works. In this paper, we match the imbalance learning methods in classification with image captioning, making the empirical study. We also propose a Task-aware Decoupled Learning and Fusion (TDLF) approach, which outperforms the former. Image captioning differs from classification in three main aspects: 1) captions are sequential labels that exist co-occurrence, 2) the generation methods usually follow the autoregressive manner, 3) the imbalance ratio is extremely large. To deal with these problems, our TDLF method introduces multi-task learning into the re-balancing approach. The model is composed of a shared autoregressor and two task classifiers, i.e. , a conventional training classifier, and a balance-training classifier. The model is further equipped with a task-aware decoupling strategy, we propose the Task Perception Indication (TPI) to measure whether the conventional training is shifted. The balance-training classifier is trained by the biased data separately and the generations of two tasks are fused according to the TPI. Experiments on the MSCOCO database show that our model outperforms the state-of-the-art methods on generation accuracy and word diversity, demonstrating the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

We study the competition of cities at a highly urbanized stage through a spatial version of the Yule model with a fixed carrying capacity. The capacity naturally determines two phases of the model, wherein the natural growth phase, we derive and observe the finite Zipf scaling of urban size distributions and the exponential decay of population density in each city; the resource constraint phase corresponds to the stage that the population of the urban system reaches the fixed capacity; thus migration among cities rather than urbanization contributes to the population dynamics of cities. In this phase, we derive the likelihood of a switch in the ranking of the cities. Our model sheds light on the post-urbanization era's inter-city competition patterns and population dynamics. [ABSTRACT FROM AUTHOR]