City Size Distribution Research Articles

Zipf's law from scale-free geometry.

The spatial distribution of people exhibits clustering across a wide range of scales, from household (∼10(-2) km) to continental (∼10(4) km) scales. Empirical data indicate 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 symmetries of scale invariance in two spatial dimensions are intimately connected to urban sociology. We test our theory by empirically measuring the power spectrum of population density fluctuations and show that the logarithmic slope α=2.04 ± 0.09, in excellent agreement with our theoretical prediction α=2. The model enables the analytic computation of many new predictions by importing the mathematical formalism of random fields.

Rank-size distributions of Chinese cities: Macro and micro patterns

A large number of studies have been conducted to find a better fit for city rank-size distributions in different countries. Many theoretical curves have been proposed, but no consensus has been reached. This study argues for the importance of examining city rank-size distribution across different city size scales. In addition to focusing on macro patterns, this study examines the micro patterns of city rank-size distributions in China. A moving window method is developed to detect rank-size distributions of cities in different sizes incrementally. The results show that micro patterns of the actual city rank-size distributions in China are much more complex than those suggested by the three theoretical distributions examined (Pareto, quadratic, and q-exponential distributions). City size distributions present persistent discontinuities. Large cities are more evenly distributed than small cities and than that predicted by Zipf’s law. In addition, the trend is becoming more pronounced over time. Medium-sized cities became evenly distributed first and then unevenly distributed thereafter. The rank-size distributions of small cities are relatively consistent. While the three theoretical distributions examined in this study all have the ability to detect the overall dynamics of city rank-size distributions, the actual macro distribution may be composed of a combination of the three theoretical distributions.

The evolution of Zipf’s law indicative of city development

Zipf’s law of city-size distributions can be expressed by three types of mathematical models: one-parameter form, two-parameter form, and three-parameter form. The one-parameter and one of the two-parameter models are familiar to urban scientists. However, the three-parameter model and another type of two-parameter model have not attracted attention. This paper is devoted to exploring the conditions and scopes of application of these Zipf models. By mathematical reasoning and empirical analysis, new discoveries are made as follows. First, if the size distribution of cities in a geographical region cannot be described with the one- or two-parameter model, maybe it can be characterized by the three-parameter model with a scaling factor and a scale-translational factor. Second, all these Zipf models can be unified by hierarchical scaling laws based on cascade structure. Third, the patterns of city-size distributions seem to evolve from three-parameter mode to two-parameter mode, and then to one-parameter mode. Four-year census data of Chinese cities are employed to verify the three-parameter Zipf’s law and the corresponding hierarchical structure of rank-size distributions. This study is revealing for people to understand the scientific laws of social systems and the property of urban development.

The power law distribution for lower tail cities in India

The city size distribution for lower tail cities has received scant attention because a small portion of the population lives in rural villages, particularly in developed countries, and data are not readily available for small cities. However, in developing countries much of the population inhabits rural areas. The purpose of this study is to test whether power law holds for small cities in India by using the most recent and comprehensive Indian census data for the year 2011. Our results show that lower tail cities for India do exhibit a power law.

Allometric scaling, size distribution and pattern formation of natural cities

AbstractThere has been a strong interest in more cogent definitions on economies of scale to reveal general urban growth laws and to develop urban performance metrics. Unstructured data, including satellite images, will provide us with new sources to do so by defining cities as aggregates of human activities. Such a uniform definition on the basis of nighttime light clusters is more consistent and economically meaningful than administrative or official boundaries. In comparison with patterns of city systems found in traditional census data, we examined the allometric scaling, size distribution and fractal geometry of natural cities. It can be concluded from our empirical analysis on regional, country and continental scales that a super-linear scaling between lightness and area with a stable exponent across different low light threshold levels generally holds for natural cities. But Zipf’s Law does not always apply over the whole range of lightness thresholds. Furthermore, we build a model based on the simple geometric matching mechanism to reproduce the self-organized formation process of nighttime light patterns. The statistical properties including allometries, size distributions and fractal geometries generated by our model are in good agreement with empirical evidence. These findings have profound implications for understanding the effects of simple aggregation behaviour in primitive stages of city formation and the urbanization process.

PARETO OR LOG‐NORMAL? BEST FIT AND TRUNCATION IN THE DISTRIBUTION OF ALL CITIES*

ABSTRACTIn the literature, the distribution of city size is a controversial issue with two common contenders: the Pareto and the log‐normal. While the first is most accredited when the distribution is truncated above a certain threshold, the latter is usually considered a better representation for the untruncated distribution of all cities. In this paper, we reassess the empirical evidence on the best‐fitting distribution in relation to the truncation point issue. Specifically, we provide a comparison among four recently proposed approaches and alternative definitions of U.S. cities. Our results highlight the importance to look at issue of the best‐fitting distribution together with the truncation issue and provide guidance with respect to the existing tests of the truncation point.

Household Registration Reform, Labor Mobility and Optimization of the Urban Hierarchy

A rational hierarchy of rank-size distribution is indispensable for optimizing the urban hierarchy. On the basis of the number of permanent urban residents, using rank-size rule testing and core density estimation within the framework of spatial economics, we set up a labor force location choice model and conducted quantitative simulations. Our findings show that the structure of China’s urban hierarchy differs from the pyramid structure of Zipf’s Law, and the household registration system impedes the free movement of labor and makes the size distribution of cities deviate from Pareto optimality. Household registration (hukou) reform will be conducive to the optimization of the urban hierarchy. Optimizing the urban hierarchy should become a consistent objective in today’s development of new-type urbanization, city clusters and metropolitan areas.

Why Do Cities Grow? Insights from Nonequilibrium Thermodynamics at the Urban and Global Scales

SummaryThis forum article explores thermodynamic understanding of the growth of cities, including theoretical foundations, observations, and analysis. The general theory of nonequilibrium thermodynamics is reviewed, recognizing differences in interpretation between Prigogine and Schneider and Kay as well as discussing the hypothesis of maximum entropy production. Calculations of exergy gradients in a few cities and settlements, along with measures of anthropogenic heat loss in further cities, support the notion that cities are dissipative structures. The observation that primary energy use per capita increases in Singapore and Hong Kong as they grow is further evidence to support the thermodynamic understanding of the growth of cities, indicative of an increasing rate of entropy production. At the global scale, the strong linear relationship between global urban population and total global energy use, and the distribution of city sizes according to Zipf's law, can be understood as emergent results based on thermodynamics. Parallel results might be derived from models that represent underlying microscale processes, several of which are reviewed. Issues for future research include: development of nonequilibrium thermodynamic models specific to city growth; further study of exergy flows of cities with consistent methodology, including attention to solar energy exchanges in cities; and further exploration of links between thermodynamic and economic models of urban growth.

From sequential to parallel growth of cities: Theory and evidence from Canada

This paper examines city growth patterns and the corresponding city size distribution evolution over long periods of time using a simple New Economic Geography (NEG) model and urban population data from Canada. The main findings are twofold. First, there is a transition from sequential to parallel growth of cities over long periods of time: city growth shows a sequential mode in the stage of rapid urbanization, i.e., the cities with the best development conditions will take the lead in growth, after which the cities with higher ranks will become the fastest-growing cities; in the late stage of urbanization, city growth converges according to Gibrat′s law, and exhibits a parallel growth pattern. Second, city size distribution is found to have persistent structural characteristics: the city system is self-organized into multiple discrete size groups; city growth shows club convergence characteristics, and the cities with similar development conditions eventually converge to a similar size. The results will not only enhance our understanding of urbanization process, but will also provide a timely and clear policy reference for promoting the healthy urbanization of developing countries.

Zipf’s law for all the natural cities around the world

Two fundamental issues surrounding research on Zipf’s law regarding city sizes are whether and why this law holds. This paper does not deal with the latter issue with respect to why, and instead investigates whether Zipf’s law holds in a global setting, thus involving all cities around the world. Unlike previous studies, which have mainly relied on conventional census data such as populations and census-bureau-imposed definitions of cities, we adopt naturally (in terms of data speak for itself) delineated cities, or natural cities, to be more precise, in order to examine Zipf’s law. We find that Zipf’s law holds remarkably well for all natural cities at the global level, and it remains almost valid at the continental level except for Africa at certain time instants. We further examine the law at the country level, and note that Zipf’s law is violated from country to country or from time to time. This violation is mainly due to our limitations; we are limited to individual countries, or to a static view on city-size distributions. The central argument of this paper is that Zipf’s law is universal, and we therefore must use the correct scope in order to observe it. We further find Zipf’s law applied to city numbers; the number of cities in the largest country is twice as many as that in the second largest country, three times as many as that in the third largest country, and so on. These findings have profound implications for big data and the science of cities.

Explaining the size distribution of cities: Extreme economies

The empirical regularity known as Zipf's law or the rank-size rule has motivated development of a theoretical literature to explain it. We examine the assumptions on consumer behavior, particularly about their inability to insure against the city-level productivity shocks, implicitly used in this literature. With either self-insurance or insurance markets, and either an arbitrarily small cost of moving or the assumption that consumers do not perfectly observe the shocks to firms' technologies, the agents will never move. Even without these frictions, our analysis yields another equilibrium with insurance where consumers never move. Thus, insurance is a substitute for movement. We propose an alternative class of models, involving extreme risk against which consumers will not insure. Instead, they will move, generating a Fréchet distribution of city sizes that is empirically competitive with other models.

On the parametric description of the French, German, Italian and Spanish city size distributions

We study the parametric description of the city size distribution of four European countries: France, Germany, Italy and Spain. The parametric models used are the lognormal, the double Pareto lognormal, the normal-Box–Cox and the threshold double Pareto Singh–Maddala (last two of these are defined in this paper). The results are quite regular. The preferred model is always the threshold double Pareto Singh–Maddala in the four countries. However, the dPln is not rejected always for the case of France, and in the case of Italy, the dPln is the runner-up distribution.

Scale-free distribution as an economic invariant: a theoretical approach

The presence of power laws (scale-free distributions) in widely different economic and social phenomena is well established. Here we focus on three specific cases viz. wealth distribution, firm size distribution and the city size distribution. We present a common framework to explain the origin of this feature in such seemingly unrelated contexts. It is shown that the equilibrium configurations of some general economic mechanisms are consistent with a power law in general and Zipf’s law in particular, in size distribution and it is an attractor under some conditions.

Evolution of Regional City Size Distribution in an Indian State: Kerala (1951-2001) Using Markovian Approach

Evolution of Regional City Size Distribution in an Indian State: Kerala (1951-2001) Using Markovian Approach

The Size Distribution of Employment Centers within the US Metropolitan Areas

This study tackles the description of the size distribution of urban employment centers or, in other words, the size of areas within cities with significantly high densities of workers. Certainly, there exists a branch of urban economics that has paid substantial attention to urban employment centers, but the efforts have been focused on identification methodologies. In this paper we build on such body of research and combine it with insights from the latest contributions in the sister subfield of city size distributions to push the agenda forward in terms of the understanding of these phenomena. We consider the 359 Metropolitan Statistical Areas (MSAs) in the United States in the year 2000 and reach three main conclusions: First, employment center sizes are more unevenly distributed than city sizes; second, the two functions that best describe city size distributions, namely the lognormal and the double Pareto-lognormal, also offer a good fit for the case of centers, particularly the latter; and third, several interesting statistically significant relationships (correlations) between variables related to centers and MSAs are deduced. Further experiments with a different technique of center identification suggest that the results are fairly robust to the method of choice.

Zipf's law in city size from a resource utilization model.

We study a resource utilization scenario characterized by intrinsic fitness. To describe the growth and organization of different cities, we consider a model for resource utilization where many restaurants compete, as in a game, to attract customers using an iterative learning process. Results for the case of restaurants with uniform fitness are reported. When fitness is uniformly distributed, it gives rise to a Zipf law for the number of customers. We perform an exact calculation for the utilization fraction for the case when choices are made independent of fitness. A variant of the model is also introduced where the fitness can be treated as an ability to stay in the business. When a restaurant loses customers, its fitness is replaced by a random fitness. The steady state fitness distribution is characterized by a power law, while the distribution of the number of customers still follows the Zipf law, implying the robustness of the model. Our model serves as a paradigm for the emergence of Zipf law in city size distribution.

Sequential city growth: Theory and evidence from the US

City growth patterns are attracting more attention in urban geography studies. This paper examines how cities develop and grow in the upper tail of size distribution in a large-scale economy based on a theoretical model under new economic geography framework and the empirical evidence from the US. The results show that cities grow in a sequential pattern. Cities with the best economic conditions are the first to grow fastest until they reach a critical size, then their growth rates slow down and the smaller cities farther down in the urban hierarchy become the fastest-growing ones in sequence. This paper also reveals three related features of urban system. First, the city size distribution evolves from low-level balanced to primate and finally high-level balanced pattern in an inverted U-shaped path. Second, there exist persistent discontinuities, or gaps, between city size classes. Third, city size in the upper tail exhibits conditional convergence characteristics. This paper could not only contribute to enhancing the understanding of urbanization process and city size distribution dynamics, but also be widely used in making effective policies and scientific urban planning.

The impact of the American Civil War on city growth

This paper analyses the persistence of the shock caused by the American Civil War on the relative city size distribution of the USA. Two features make the study of this conflict interesting. First, it took place at an earlier stage of the industrialisation and urbanisation processes than those previously analysed in the related literature. Second, the battles were fought in the open field, not in urban areas. In line with previous results for the Second World War in Japanese and German cities, our findings suggest that the effects of the shock were transitory. Furthermore, some evidence regarding the possible presence of a ‘safe harbour effect’ is reported.

THE TEMPORAL AND SPATIAL DYNAMICS OF REGIONAL CITY-SIZE DISTRIBUTION: AN INDIAN STATE -- KERALA (1951-2001)

This paper aims at evaluating the temporal and spatial dynamics of City Size Distribution in Kerala, an Indian State for the period 1951-2001.The study is based on a function relating population size to rank and the same is used to test for the trends of deconcentration and spatial decentralization.The expansion methodology is used to investigate the dynamics of rank size function in both temporal and spatial dimensions.Our findings reveals that the size distribution of cities/towns in the state shows an increasing and decreasing pattern and indicating a more even and less even distribution cities/towns populations.The study also indicates that distance from a central city is not a significant factor in any trend towards population deconcentration.

City age and city size

Using novel data on the foundation dates of more than 10,000 American Census places, we show that older cities in the US tend to be larger than younger ones. To take this nexus between city age and city size into account, we introduce endogenous city creation into a dynamic economic model of an urban system. Our model predicts a pattern of age-dependent urban growth that is in line with recently established empirical evidence. The size distribution that emerges in our economy delivers a close fit to different types of US data and it outperforms other suggested parametrizations. This evidence can resolve several debates and build a bridge between different views in the literature on city size distributions.