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Open Access
Peer-reviewed
Research Article
- Andrew J. Kerkhoff,
- Melanie E. Moses
Global Patterns of City Size Distributions and Their Fundamental Drivers
- Ethan H. Decker,
- Andrew J. Kerkhoff,
- Melanie E. Moses
x
- Published: September 26, 2007
- //doi.org/10.1371/journal.pone.0000934
Figures
Abstract
Urban areas and their voracious appetites are increasingly dominating the flows of energy and materials around the globe. Understanding the size distribution and dynamics of urban areas is vital if we are to manage their growth and mitigate their negative impacts on global ecosystems. For over 50 years, city size distributions have been assumed to universally follow a power function, and many theories have been put forth to explain what has become known as Zipf's law [the instance where the exponent of the power function equals unity]. Most previous studies, however, only include the largest cities that comprise the tail of the distribution. Here we show that national, regional and continental city size distributions, whether based on census data or inferred from cluster areas of remotely-sensed nighttime lights, are in fact lognormally distributed through the majority of cities and only approach power functions for the largest cities in the distribution tails. To explore generating processes, we use a simple model incorporating only two basic human dynamics, migration and reproduction, that nonetheless generates distributions very similar to those found empirically. Our results suggest that macroscopic patterns of human settlements may be far more constrained by fundamental ecological principles than more fine-scale socioeconomic factors.
Citation: Decker EH, Kerkhoff AJ, Moses ME [2007] Global Patterns of City Size Distributions and Their Fundamental Drivers. PLoS ONE 2[9]: e934. //doi.org/10.1371/journal.pone.0000934
Academic Editor: Jerome Chave, Centre National de la Recherche Scientifique, France
Received: January 21, 2007; Accepted: September 5, 2007; Published: September 26, 2007
Copyright: © 2007 Decker et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Funding: This research was supported in part by NASA and a National Science Foundation Doctoral Research Fellowship. Both groups funded the research but were not involved in conducting or analyzing the research.
Competing interests: The authors have declared that no competing interests exist.
Introduction
Humans increasingly dominate the ecology and energy flows of the entire earth, prompting grave concerns about human population growth. However, the human population has not only doubled in the past 40 years, but that population is increasingly clustered in urban areas. In 1950, only 30% of the world's population lived in urban areas. By 2000 that proportion rose to 47%, and by 2030 that number will be 60%[1]. In fact, virtually all of the global population growth in the next 25 years will be urban, either through migration from rural areas, growth of existing cities, or the emergence of new urban clusters. In less developed countries, cities are burdened by the growth of unregulated slums, illegal or unmanaged waste and sewage disposal, and woefully inadequate water supplies, housing, and transportation infrastructure [2]–[4]. In more developed regions, rapid urban sprawl and the growth of the built-up urban fringe have outpaced much of the environmental and urban planning that attempt to manage them [3]–[6].
The increasingly global ramifications of human urbanization necessitate a global perspective on the problem. If we hope to successfully manage urban environmental impacts, we first need to know how urban areas are distributed and how that distribution varies around the world. Second, we need to understand what basic ecological principles [if any] underlie that distribution and how those principles embody themselves in human behavior. Finally, we need to understand how the per capita environmental impact of humans varies across settlements of different sizes and across regions that differ economically, culturally, and biogeographically. This paper addresses the first two points by quantifying the size distribution of urban areas around the world and modeling their ecological bases in the dynamics of human migration and reproduction.
Urbanization is occurring so quickly in many areas that it has become difficult to distinguish city, suburb, and town. There are many ways to define a city, e.g., as an incorporated area, an urban agglomeration, or a settlement with population density larger than some threshold value. All definitions have shortcomings: is Newark part of the New York City metropolitan area? Is Santa Fe, New Mexico, a city or a town? What about an urban area that straddles a county, state, or even national border, such as Kansas City or El Paso–Ciudad Juarez? For this paper we use the term city very loosely to mean any human settlement that is functionally coherent and denser than its surroundings, and we use it interchangeably with the term settlement. We do not distinguish here between villages, towns, cities, metropolitan areas, and megacities [though differences of kind certainly exist]. Generally, we are interested in understanding the flow of energy and materials through human networks, and as virtually all commerce requires some aggregation of population to occur, we are interested in the entire set of human settlements.
Current theories of city size distributions
Consider a set of cities from a region, such as a country, ranked by population [or by area] from largest to smallest. When population is graphed against rank, the shape of the curve describes the relative proportions of smaller and larger cities. In 1949, Zipf observed that the population of a city is proportional to the inverse of its regional rank [7], resulting in a power law that has an exponent of approximately −1. Equivalently, this observation, which has become known as “Zipf's law,” states that the probability that the size of a city s is greater than some S is proportional to 1/S: P[s>S] = cS x. with x = −1.
The mechanisms underlying Zipf's law have been the subject of much theoretical debate [8]–[12]. Because pattern and process are intricately linked in natural phenomena [13]–[15], governing processes are often inferred from the observed patterns [16]; power laws are often taken as evidence that these processes are scale-invariant [17], [18]. Several researchers have hypothesized that Zipf's law is a result of all cities growing at the same rate, regardless of their size. This law of proportionate effect is also known as Gibrat's law [19]. Others suggest that a steady rate of new cities joining an urban system [i.e., Yule's theorem] produces the power function [8]. In fact, there is a wide variety of theories for Zipf's law, ranging from the statistical-mechanical to the sociological and political [discussed in Andersson 2002 [20], and Ioannides&Gabaix [21]]. A few regional studies have suggested that a lognormal best describes the city size distribution [21], particularly when the smallest cities are included [22], [23]. Carroll [24] reviews much of the early literature on Zipf's law.
What almost all explanations have in common is the assumption that Zipf's law is a robust empirical pattern that requires explanation. However, we argue that insufficient consideration has been given to 1] testing the applicability of Zipf's law over the entire range of human settlement sizes, and 2] developing useful “neutral models” for understanding the extent to which settlement distributions represent stochastic vs. deterministic, goal-directed [e.g., optimization] processes. These two points are critical to assessing the generality and meaningfulness of Zipf's law, i.e., whether it actually teaches us anything about human ecology and the organization of human populations. Even on a more practical level, we cannot apply Zipf's law as even an empirical descriptor of the distribution of human settlements without more fully addressing its generality.
Methods
Empirical settlement size distributions
Most studies of city size distributions have concentrated on only the largest cities and have ignored smaller cities, towns, and settlements, mainly because suitably accurate data for small cities did not exist. Yet as much as 70% of the population may reside in these smaller areas; omitting that mass of the population may lead to biased characterizations of city size distributions.
For many regions around the globe, large cities do follow power functions. Figure 1 shows the rank-size distributions and power-law fits of population P to rank R for the largest cities from three data sets: metropolises of the world [25] [P = 5.9×107 R−0.686, standard error of exponent = 0.006, r2 = 0.847, p1; 3] cluster populations, summed across all cells in a cluster; or 4] cluster area measured as the number of cells in a cluster. These correspond to different measurement criteria for real cities: all cities, large cities only [the tail of the distribution], metropolitan area population and metropolitan area extent.
Figure 3. Development of the urban growth simulation, shown here on a small lattice of L = 32.
a] At time t = 0 there is some population n[0] in each cell. b] Time t = 4. c] Time t = 8. d] By time t = 16, intermittent spikes of large population are clustered together in an otherwise sparsely-populated lattice.
//doi.org/10.1371/journal.pone.0000934.g003
Results
Empirical results
For both the census data [Figure 4] and the night light clusters [Figure 5], the tails of all distributions [e.g., light cluster areas exceeding 50 km2] appear to approximate power laws [using maximum likelihood estimation]. However, slopes of power laws [fitted over the large cities only] are almost all significantly different from each other and lie between −0.729 and −0.888, far from the theoretically expected Zipf exponent of −1.
Figure 4. Rank-size distributions and lognormal fits [dashed blue lines] for all ‘counties’ [2 administrative levels below the nation] of the world, all populated places of the USA, and all municipalities of Switzerland.
All three samples are fit well by a lognormal over four orders of magnitude in population, particularly in the bodies of the distributions. Key indicates sample sizes.
//doi.org/10.1371/journal.pone.0000934.g004
Figure 5. Cumulative probability distributions of nighttime light clusters for continental and sub-continental regions.
Key indicates number of clusters in each region [total N = 68,530]. The body of each distribution is fit well by a lognormal [not shown]. The tails [largest clusters] are fit well by power laws [not shown]. Slopes of the scaling regions are all significantly different from the Zipf's law expectation of −1.
//doi.org/10.1371/journal.pone.0000934.g005
More remarkably, when all city sizes are considered, the bulk of the data are better fit by a lognormal distribution than by a power law. Figure 4 shows the rank-size distributions for the same locations as Figures 1 and 2, but this time including all settlements, which increases the sample size ten-fold for Swiss municipalities, 58-fold for World ‘counties’ [2 administrative levels below the nation], and 75-fold for US populated places. Maximum likelihood estimates of the lognormal parameters [μ and σ] for of the world [μ = 10.95, σ = 1.92, r2 = 0.439, p