City Size Distribution Research Articles

A test of the relationship between the Pareto exponent and sample size

This paper uses un-truncated city population data from three countries—the United States, Spain and Italy—to empirically test Proposition 1 put forth by Eeckhout (2004 American Economic Review, 94: 1429–1451). Eeckhout’s hypothesis was that the estimate of the Pareto exponent in a standard Zipf regression decreases with sample size, if the underlying city size distribution is lognormal. Using rolling sample regressions, we find that this proposition is only valid once we enter the lognormal body of the distribution; for the Pareto-distributed upper-tail, the estimated exponent does not vary with sample size.

More urbanization, more polarization: evidence from two decades of urban expansion in China

This article examines whether the patterns of urbanization in China over the past twenty years have resulted in a more balanced or increasingly polarized city size distribution. China is an important study area due to the immense scale of its urbanization and the substantial role of its planners in coordinating and striving for more balanced development patterns. This study utilizes measures of the spatial and temporal characteristics of built-up area (BUA) expansion across 23 urban agglomerations in China, focusing on size dynamics through the urban primacy index and Pareto exponent. The BUA within these 23 UAs increased by 280.90% in the study period. Coastal urban agglomerations were larger in size at baseline and maintained relatively high annual growth rates compared to inland agglomerations, indicating rising polarization. Changes within agglomerations were indicative of declining polarization, as evidenced by the reduction in the urban primacy index and the increase of Pareto exponent. Policies implemented to achieve enhanced urbanization coordination in China appear to be partially successful, but the growth of urban regions with existing large size continues to fuel uneven urban development and polarization.

Granular Cities

This study extends the concept of granularity from firms to cities, examining how large cities influence national economic dynamics beyond their relative size. By applying Zipf’s law, which describes the power law distribution of city sizes, we investigate the interplay between granularity and business cycles. Our aim is to test the granular hypothesis that large cities have a significant impact on the business cycle beyond their relative size. We analyze data from American and Brazilian cities between 2003 and 2019 assessing the granular residuals and their explanatory power. Our findings reveal that in the United States, the granular city size is three metropolitan areas or five counties when redefined. In Brazil, it equates to three municipalities. These results emphasize the substantial role large cities play in national economic fluctuations, suggesting that policy interventions that target infrastructure, education, and innovation in major urban centers could have widespread economic benefits. This paper’s contribution to the literature is to highlight a spatial component of granularity not considered so far.

City size distribution of towns in the 17th century in Lower Silesia

AbstractThis paper attempts to verify the hierarchy of towns in Lower Silesia on the basis of population data from the second decade of the 17th century, assuming that a statistically described distribution law applies. The analysis of the city size distribution was based on Gibrat's law and Zipf's rank‐size rule. This made it possible to verify the assumptions made at the outset and to estimate the population of towns of unknown size and designated rank. The research carried out suggests a revision of the estimates, especially for the largest cities in the region, which tend to be underestimated. Studies have also shown that the method can be extremely useful for verifying uncertain and incomplete historical demographic data.

The Delineation and Characterisation of Chinese Urbanised Areas Using Micro-Scale Population Census Data

The definition of cities from the physical view in China is lacking. Most research regarding Chinese cities relies on the definition from the administrative view, which is often inconsistent as there are serious discrepancies between the administrative boundaries and the physical extent of Chinese cities. This study aims to delineate Chinese urbanised areas using population census data at the township level and analyse the identified urbanised areas’ characteristics. The results show that the numbers of Chinese urbanised areas in 2000, 2010, and 2020 were 758, 942, and 1058, respectively, indicating there are several ‘cities outside the system’ that meet the requirements of urban agglomeration but are constrained by the city designation system. The degree of population aggregation and the growth rate of urbanised areas are greater than those of administrative cities. This indicates that the geographical scopes of administrative cities cannot accurately reflect the ‘real’ urban areas. Additionally, the city-size distribution follows Zipf’s law, with the Zipf coefficient moving closer to one if we fit the Zipf model based on the urbanised areas. It is necessary to establish official urbanised areas and publish statistical data based on urbanised areas, which could have significant implications both for policymakers and researchers.

Comparing city size distributions: Gridded population versus nighttime lights

AbstractThis paper compares the size distributions of cities when they are measured using gridded population and nighttime lights (NTLs) data. To do so, we exploit recent and accurate satellite imagery to proxy urban economic activity. Similarly to related studies, our results suggest that population is more equally distributed than lights at the country level. However, and calling assumptions established for urban NTLs into question, our findings do not support a Pareto function for their distribution. We also obtain evidence of a nonlinear and heterogeneous link between population and lights for a global sample of cities. Grounded on our empirical analysis, we develop a simple theoretical framework that relates the difference between the distributions of population and light emissions to the strength of agglomeration economies.

A long-term, regional-level analysis of Zipf's and Gibrat's laws in the United States

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.

How to define the city size in China? A review over a century from 1918 to 2020

Accurately defining the size of cities is essential for urban studies. As the country with the world's largest urban population, China has experienced numerous changes in its definition of city size. We carefully collected and analyzed China's city size based on authoritative population census or recorded data from 1918 to 2020. Our key findings are as follows. (1) Over the past century, the administrative proper of Chinese cities has seen a marked expansion, from its proximity to urban entities to the concept of an extended region. The definition of city size in China was optimized from the total population within the city proper based on the administrative boundary to that within the city entity based on the urbanized built-up area. (2) There are two main aspects to optimizing China's definition of city size. The first is the use of township- or village-level administrative units to match the city entities. The second is the use of residential population rather than the traditional hukou registration population to indicate size. (3) China's city size distribution based on the city entity has shifted from being predominantly small- and medium-sized cities to being dominated by large cities. Cities with larger populations a century ago, mainly located in coastal urban agglomerations and inland provincial capitals, still had a strong size advantage a century later. The experiences of optimizing the definition of city size in China offer implications for other rapidly urbanizing countries and for international city size comparisons.

Auerbach, Lotka, and Zipf: pioneers of power-law city-size distributions

Power-law city-size distributions are a statistical regularity researched in many countries and urban systems. In this history of science treatise we reconsider Felix Auerbach’s paper published in 1913. We reviewed his analysis and found (i) that a constant absolute concentration, as introduced by him, is equivalent to a power-law distribution with exponent approx 1, (ii) that Auerbach describes this equivalence, and (iii) that Auerbach also pioneered the empirical analysis of city-size distributions across countries, regions, and time periods. We further investigate his legacy as reflected in citations and find that important follow-up work, e.g. by Lotka (Elements of physical biology. Williams & Wilkins Company, Baltimore, 1925) and Zipf (Human behavior and the principle of least effort: an introduction to human ecology, Martino Publishing, Manfield Centre, CT (2012), 1949), does give proper reference to his discovery—but others do not. For example, only approximately 20% of city-related works citing Zipf (1949) also cite Auerbach (Petermanns Geogr Mitteilungen 59(74):74–76, 1913). To our best knowledge, Lotka (1925) was the first to describe the power-law rank-size rule as it is analyzed today. Saibante (Metron Rivista Internazionale di Statistica 7(2):53–99, 1928), building on Auerbach and Lotka, investigated the power-law rank-size rule across countries, regions, and time periods. Zipf’s achievement was to embed these findings in his monumental 1949 book. We suggest that the use of “Auerbach–Lotka–Zipf law” (or “ALZ-law”) is more appropriate than “Zipf’s law for cities”, which also avoids confusion with Zipf’s law for word frequency. We end the treatise with biographical notes on Auerbach.

Exciting, boring, and nonexistent skylines: Vertical building gaps in global perspective

AbstractDespite the widespread prevalence and economic importance of tall buildings, little is known about how their patterns vary across space and time. We aim to quantify differences across major world regions over time (1950–2020). To do so, we exploit novel data on the location, height (above 55 m), and year of construction of nearly all tall buildings in the world. We propose a new methodology to estimate the extent to which some world regions build up more than others given similar economic and geographic conditions, city size distributions, and other features. Our analyses reveal that many skylines may visually appear more prominent than they really are once one includes all tall buildings and core controls, which alters how regions are ranked in terms of tall building stocks. Using results by city size, centrality, height of buildings, and building function, we classify world regions into different groups, finding that tall building stocks are likely driven by boring skylines of residential high‐rises, and to a lesser extent exciting skylines of skyscrapers and office towers. Finally, land‐use regulations and preferences, not historical preservation nor dispersed ownership, may account for most observed differences.

Intra city-size distribution in the Yangtze River Delta Region: Equalization or polarization

In the context of China’s New Urbanization Strategy, a growing number of cities are beginning to experience significant changes in their intra-city structure and size distribution, but systematic research is still lacking. This paper investigated the intra city-size distribution of 214 cities in China’s Yangtze River Delta Region (YRDR) from 1990 to 2020. The study used the Pareto index to portray the equalization differences in intra city-size distribution, and investigated the heterogeneity of its influencing factors using multiscale geographically weighted regression (MGWR). The results showed that most of the cities in the YRDR had a city-size distribution consistent with a Pareto distribution but still tended to be polarized. Among them, the better-developed large cities and regions were more consistent with the Zipf distribution, as were the county-level cities and prefecture-level cities. In terms of spatio-temporal distribution, the intra city-size distribution showed a spatial agglomeration pattern of “polarization in the west and equalization in the east” and the scope and intensity of cold and hot spots reduced. Regarding driving factors, land, industry, and population were essential influences on the intra city-size distribution. Of these, the natural environment and location were important global influencing factors, and economic and social factors only had a more significant influence in 1990 and 2020. The impact of industry restructuring gradually increased, and the population was a significant influence only in the early years. By exploring the intra city-size distribution, this paper provides a scientific basis for the optimization of town structure and policy formulation in small and medium-sized cities, and some new concerns have been proposed.

Scaling in city size distributions

Scaling in city size distributions

Spatial and Temporal Evolution Characteristics of China’s City Size Distribution Based on New Criteria

The distribution and evolution of city size are critical for town layout optimization. Based on the most recent classification standards and census data for 2010 and 2020, this paper aims to explore China’s city size distribution above the prefecture level. Using the rank-size law, Kernel density estimation, Spatial Gini coefficient, and Markov transition matrix, the newest city size distribution characteristics and spatial evolution patterns in China are shown from national and regional viewpoints. Our main findings are as follows: (1) Over the period from 2010 to 2020, China’s city size distribution follows the rank-size law but deviates from Zipf’s ideal. The distribution of city size is centralized in general. (2) China’s city-size hierarchy exhibits a good “olive” structure, with fewer megacities but larger populations. The growth rate of small and medium-sized cities is higher than the number of medium-sized cities. (3) China’s cities have grown greatly in size, with more than a third of them expanding. Over the last decade, high-ranking cities have become the primary driver of change. (4) There are disparities in city size between regions. A diminishing trend can be seen in three key economic zones.

Discontinuous structure of regional and subregional urban systems: Nouvelle-Aquitaine, France (1800–2015)

In the prevailing literature in urban economics, conducting research on urban systems at the national scale tends to provide an elegant but reductive approach to the functioning of these systems, assuming, in particular, that city size distributions are continuous. Based on an alternative framework drawing from research in ecology and complex adaptive systems, this article questions this approach by testing the discontinuity hypothesis within regional urban systems in France using two methods and long-term census data (1800–2015). We found that the distribution of city sizes over the 200+ years of population data from the Nouvelle-Aquitaine region was discontinuous for every year of data. The distributions consist of groups of similarly sized cities, separated by gaps (discontinuities) where there are no cities at all. The location and number of discontinuities were conservative over time, suggesting that the processes shaping size classes are conservative and largely independent of societal disturbances such as wars. The two methods used to identify discontinuities were highly congruent. Finally, the Nouvelle-Aquitaine region is comprised of three former regions, Aquitaine, Limousin and Poitou-Charentes, each of which also had discontinuous city size distributions with conservative structures over time. The study results question the traditional expectations about the growth and development of urban systems.

The Current State of Territorial Development of ASEAN Countries and Strategic Types for Balanced Development

ASEAN countries are promoting regionally balanced development in their policy in response to development disparity resulting from rapid urbanization. This study aims to draw strategic approaches for balanced development based on the current status of territorial development of the ASEAN countries. Through the lens of Korea’s territorial development experience, this study presents three strategic types for balanced territorial development and an analysis framework to connect the strategic types and the territorial conditions of ASEAN countries. This study analyzes the current status from five aspects: (1) the current status of urbanization and industrialization, (2) the size distribution of cities, (3) the spatial structure of the territories, (4) the state of regional development, and (5) balanced development visions and territorial policies. Based on the analysis results, this study categorizes the ASEAN countries that exhibit similar territorial development and applies the strategic types. This study derives policy measures suitable to each strategic type that will help the ASEAN countries to facilitate balanced development in accordance with their territorial state.

City Size Distribution Analyses Based on the Concept of Entropy Competition

The present work pursues theoretical and empirical objectives. With regards to the former, it is demonstrated that the natural tendency to uniformity of both the probability distribution of a city to have a certain number of inhabitants and that of a person to reside in a town of a given number of citizens leads to a competition between their information entropies, which provides the power law distribution as the most probable one for city size. It is also shown that Zipf's law reflects the significant control of the existence of interconnections between cities on the self‐organization of their size. With regards to the empirical objectives, based on population data of European countries and Italian municipalities, the theoretical approach proposed is validated. At the Italian scale, city distribution is shown to be a power law for cities above 10,000 inhabitants. In the 20 Italian regions, the breakpoint in the distribution is generally lower. Finally, the geographical control on city distribution is discussed based on the results achieved in some regions.

New economy and national city size distribution

This paper examines the effects of the new economy on national city size distribution in 102 countries. Results show that the new economy has significant effects on city size distributions and such effects are heterogeneous across countries. Human capital contributes to the polarization of city size distribution in developed countries and service-dominated countries, while it helps to equalize city size distribution in industrial-dominated countries. While innovation promotes an equalized city size distribution in general, globalization promotes polarization and then equalization of city size distribution. Information and communication technology triggers an equalized followed by a polarized city size distribution, and has contributed to an equalized city size distribution in service-dominated countries. It is evident that thriving new economic processes are changing the global urban hierarchy and reshaping national city size distribution. This study facilitates an in-depth understanding of the effects of the new economy on city size distribution, which is of great importance to promote more equitable development and urbanization.

Monitoring the Distribution and Variations of City Size Based on Night-Time Light Remote Sensing: A Case Study in the Yangtze River Delta of China

Effectively monitoring the size of a city in real time enables the scientific planning of urban development. Models that utilize the distribution and variations in city size generally use population data as inputs, which cannot be obtained in a timely and rapid manner. However, night-time light (NTL) remote sensing may be an alternative method. A case study was carried out on the Yangtze River Delta (YRD) in China, and the rank–size rule, the law of primate cities, and the Gini coefficient were employed to monitor the variation in city size in the study area. The urban areas extracted based on NTL remote sensing were utilized instead of the traditionally used population data to evaluate the variations in city size from 2012 to 2017. Considering the empiricism and subjectivity of the thresholding method, urban areas were extracted from NTL data combined with the normalized differential vegetation index and land-surface temperature data based on the artificial neural network algorithm. Based on the results, the YRD did not fit the distribution of the primate cities from 2012 to 2017. However, this region satisfied the rank–size rule well, which indicated that the development of medium–small cities was more prominent than that of larger cities, and the dispersed force was larger than the concentrated force. Notably, the city size reached a relatively balanced level in the study area. Further, sensitivity analysis revealed that the relatively low extraction accuracy of urban areas of few small cities had little effect on the results of city size variations. Moreover, the validation of city size computed from statistical population data and its comparison with results calculated based on the statistical data of urban areas aligned with the results of this study, which indicates the rationality and applicability of monitoring the variations in city size using the urban areas extracted from NTL remote sensing instead of population data.

Theoretical study of the effect of ports in the formation of city systems

This paper explores theoretically the formation of a system of cities in which ports affect the spatial location and the size of cities. We use a complex systems and economic geography approach to generate 2D cellular automata to simulate the formation of the landscape of urban agglomerations based on different configurations of port locations. The dynamics of the model shows the emergence of the classical city-size distributions in which the number of ports and their layout affects the growth rate and location of the city-size values. Our findings showed that the two port-city configuration give rise to cities at a long distance from a small number of ports. The size of the cities shows a positive correlations with their distance to ports. A four port-city configuration showed that if the number of ports increases, lower city-sizes are attained and their population displays a negative correlations with their distance to ports. For a lateral configuration with a significantly increase in the number of ports, mainly homogeneous city-size distributions are favored with a slight long-distance size correlation. Therefore, our theoretical model shows a high internal consistency between the theory and assumptions used for describing reliable scenarios in the relationship of ports and urban systems.

Diversity in global urban sprawl patterns revealed by Zipfian dynamics

ABSTRACT Zipf’s law, characterizing the rank-size distribution of city size or population, has been widely applied in urban studies. Previous studies on Zipf’s law mainly focused on the spatial dimension of urban size and population, with limited consideration of its temporal dynamics. Here, we proposed a conceptual model to characterize the spatial sprawl pattern of urban clusters using the approach of Zipf’s law and approximately 30-year time series of global urban extent data. First, we quantified increments of urban areas in global megacities and small settlements over the past three decades. Urban sprawl patterns (i.e., equilibrium, diffuse, and aggregated) were revealed from the Zipfian dynamics, using the proposed conceptual model. Then, we characterized urban sprawl patterns at different spatial scales. Overall, the sprawl of small urban clusters is slightly faster than those of large clusters globally. At the continental scale, the sprawl pattern shows equilibrium patterns in Asia and Africa, whereas other continents mainly present diffuse sprawl patterns. In general, for developing regions (e.g., North and West Africa), an aggregated sprawl pattern was observed, whereas, for those highly developed regions (e.g., Canada and West Europe), the diffuse sprawl pattern dominated. Patterns of urban sprawl revealed in this study reflect different growth rates of varying sizes of urban clusters, advancing our understanding of urban development pathways. The relatively robust performance of Zipf’s law over multiple spatial and temporal dimensions suggests the self-organized mechanism of city evolution behind urban sprawl, which contributes to the development of future urban growth models.