This work studies the Zipf Law for cities in Brazil. Data from censuses of 1970, 1980, 1991 and 2000 were used to select a sample containing only cities with 30,000 inhabitants or more. The results show that the population distribution in Brazilian cities does follow a power law similar to the ones found in other countries. Estimates of the power law exponent were found to be 2.22 +/- 0.34 for the 1970 and 1980 censuses, and 2.26 +/- 0.11 for censuses of 1991 and 2000. More accurate results were obtained with the maximum likelihood estimator, showing an exponent equal to 2.41 for 1970 and 2.36 for the other three years.
The recent quantitative approaches for studing several aspects of urban life and infrastructure have shown that scale properties allow to understand many features of urban infrastructure and of human activity in cities. In this work, an analysis based on the complexity of cities network is performed for data on Brazilian cities firearms homicides. Due to the diversity in Brazilian population and cities, this is an interesting test for theories recently proposed. The superlinear power-law behavior for number of homicides as a function of city population, with exponent $β=1.15$ is obtained. The …
The empirical studies of city-size distribution show that Zipf's law and the hierarchical scaling law are linked in many ways. The rank-size scaling and hierarchical scaling seem to be two different sides of the same coin, but their relationship has never been revealed by strict mathematical proof. In this paper, the Zipf's distribution of cities is abstracted as a q-sequence. Based on this sequence, a self-similar hierarchy consisting of many levels is defined and the numbers of cities in different levels form a geometric sequence. An exponential distribution of the average size of cities is …
The rank-size plots of a large number of different physical and socio-economic systems are usually said to follow Zipf's law, but a unique framework for the comprehension of this ubiquitous scaling law is still lacking. Here we show that a dynamical approach is crucial: during their evolution, some systems are attracted towards Zipf's law, while others presents Zipf's law only temporarily and, therefore, spuriously. A truly Zipfian dynamics is characterized by a dynamical constraint, or coherence, among the parameters of the generating PDF, and the number of elements in the system. A clear-cut…
Power law distributions characterise several natural and social phenomena. The Zipf law for cities is one of those. The study views the question of whether that global regularity is independent of different spatial distributions of cities. For that purpose, a typical Zipfian rank-size distribution of cities is generated with random numbers. This distribution is then cast into different settings of spatial coordinates. For the estimation, the variables rank and size are supplemented by spatial spillover effects in a standard spatial econometric approach. Results suggest that distance and contig…
The rank-size distribution of cities follows Zipf's law, and the Zipf scaling exponent often tends to a constant 1. This seems to be a general rule. However, a recent numerical experiment shows that there exists a contradiction between the Zipf exponent 1 and high urbanization level in a large population country. In this paper, mathematical modeling, computational analysis, and the method of proof by contradiction are employed to reveal the numerical relationships between urbanization level and Zipf scaling exponent. The main findings are as follows. (1) If Zipf scaling exponent equals 1, the …
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 bot…
The distribution of the population of cities has attracted a great deal of attention, in part because it sharply constrains models of local growth. However, to this day, there is no consensus on the distribution below the very upper tail, because available data need to rely on the "legal" rather than "economic" definition of cities for medium and small cities. To remedy this difficulty, in this work we construct cities "from the bottom up" by clustering populated areas obtained from high-resolution data. This method allows us to investigate the population and area of cities for urban agglomera…
We propose hypotheses describing the empirical finding of an association between the exponents of urban GDP scaling and Zipf's law for cities. These hypotheses represent various combinations of directional or reciprocal causal links between the two phenomena and include inter- and intra-city processes. Future theories and models can be motivated with and categorized according to these hypotheses. This paper intends to stimulate the discussion around the processes behind these phenomena and pave the way to a Unified Urban Theory.
The law of allometric scaling based on Zipf distributions can be employed to research hierarchies of cities in a geographical region. However, the allometric patterns are easily influenced by random disturbance from the noises in observational data. In theory, both the allometric growth law and Zipf's law are related to the hierarchical scaling laws associated with fractal structure. In this paper, the scaling laws of hierarchies with cascade structure are used to study Chinese cities, and the method of R/S analysis is applied to analyzing the change trend of the allometric scaling exponents. …
Zipf's law for cities is probably the most famous regularity in social sciences. So much that, a hundred years of publication later, its status is not clear: is it a law of social organisation? Is it an instrument of description of city size distributions? Is it an element of validation of geographical objects (cities and systems of cities in particular)? Empirical estimations of the rank-size parameters are very numerous and contradict each other. In this study, we present the results of a reproducible meta-analysis of the largest pool of papers regarding this issue, obtained from the collect…
The spatial distribution of people exhibits clustering across a wide range of scales, from household ($\sim 10^{-2}$ km) to continental ($\sim 10^4$ km) scales. Empirical data indicates simple power-law scalings for the size distribution of cities (known as Zipf's law) and the population density fluctuations as a function of scale. Using techniques from random field theory and statistical physics, we show that these power laws are fundamentally a consequence of the scale-free spatial clustering of human populations and the fact that humans inhabit a two-dimensional surface. In this sense, the …
Université Paris-Saclay（巴黎萨克雷大学）；CEA（法国原子能和替代能源委员会）；CNRS（法国国家科学研究中心）；Institut de Physique Théorique（理论物理研究所）；Centre d’Analyse et de Mathématique Sociales（社会分析与数学中心）；EHESS（高等社会科学学院）
The gravity model is one of important models of social physics and human geography, but several basic theoretical and methodological problems remain to be solved. In particular, it is hard to explain and evaluate the distance exponent using the ideas from Euclidean geometry. This paper is devoted to exploring the distance-decay parameter of the urban gravity model. Based on the concepts from fractal geometry, several fractal parameter relations can be derived from scaling laws of self-similar hierarchies of cities. Results show that the distance exponent is just a scaling exponent, which equal…
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…
Empirical analysis on the connection between power-law distributions and allometries for urban indicators
我们报告现有的幂律分布和异速生长之间的联系。正如在[PLoS ONE 7，e40393（2012）]中首次报道的凶杀案与人口之间的关系，当这些城市指标呈现渐近幂律分布时，它们也可以显示它们之间的特定异速生长。在这里，我们提出了一个广泛的特性，这种连接时，考虑所有可能的关系对巴西城市的12个城市指标（如童工，文盲，收入，卫生和失业）。我们的分析表明，我们所有的城市指标是渐近分布的幂律和建议的连接也适用于我们的数据时，异速生长关系显示足够的相关性。我们还发现，并非所有的异速生长关系都是独立的，它们可以被理解为城市指标和人口规模之间的异速生长关系的结果。我们进一步表明，周围的异速生长的残差波动的特点是几乎恒定的方差和对数正态分布。
