Source-linked AI summary
Zipf's Law for All the Natural Cities around the World
Bin Jiang, Junjun Yin, Qingling Liu
TL;DR
Whether Zipf’s law holds for city sizes is often examined within individual countries, though small samples and cross-border connections complicate that scope. The paper tests the law using naturally delineated cities at global, continental, and country levels, finding its strongest fit globally and a related regularity in country city counts.
Problem
Country-by-country tests may use too few cities for a statistical regularity, while urban connections such as primate-city impacts can cross national borders.
Source: §1 · p. 2Method
The study extracts natural cities from three images using an average head/tail-breaks threshold, then tests Zipf’s law at global, continental, and country levels.
Source: §4 · p. 6Results
Zipf’s law holds remarkably well for global city sizes, remains generally valid on continents except Africa in 2010, and also describes city numbers across countries.
Source: §4 · pp. 6, 8Takeaways & Limitations
Abstract
from arXiv · showhide
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 speaks 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 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 first 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. Keywords: Night-time imagery, city-size distributions, head/tail division rule, head/tail breaks, big data
1. Introduction
The paper revisits conflicting claims about Zipf’s law by asking whether it applies to all cities worldwide, rather than only cities within individual countries. It argues that natural-city data and an appropriate global scope are needed to assess the law.
1. Introduction
- Prior studies offer a contradictory picture: Zipf’s law is described as widespread, yet reported violations and country-level differences leave its scope unsettled.The paper focuses on the law with exponent one (±0.1).Source: §1 · pp. 1–2
- A global test is motivated by small national samples and by city interactions, including primate-city effects, that can cross country borders.The paper argues that all cities worldwide form an interconnected whole and asks whether Zipf’s law applies to that scope.Source: §1 · p. 2
- The study examines natural cities from three night-time images spanning 19 years and uses maximum-likelihood estimates at global, continental, and country levels.It also reports that city numbers across countries follow Zipf’s law.Source: §1 · p. 2
2. Natural cities extracted from night-time imagery
This section describes delineating natural cities from night-time imagery using recursive head/tail breaks and a shared cutoff, while noting that city boundaries depend on threshold choices.
2. Natural cities extracted from night-time imagery
- Remote sensing can delimit cities’ extents and locations, but census-based definitions make automated extraction uncertain because city boundaries depend on how cities are defined.The section contrasts administrative definitions with naturally delineated settlements.Source: §2 · p. 2
- Natural cities are delineated by recursively splitting heavy-tailed pixel values at their mean into a head above the mean and a tail below it.The method continues splitting the head to identify a meaningful cutoff, then groups connected pixels above that cutoff as natural cities.Source: §2 · p. 3
- The study used 31 night-time images spanning 1992–2010, calibrated pixel values across years, and represented the imagery in a Mollweide equal-area projection.The imagery had approximately 30-arc-second, or 1-kilometer, resolution.Source: §2 · p. 3, Figure 1 · p. 3
- In the U.S. image example, 11,766,012 pixels split at a mean of 7.5 into 3,091,666 head pixels (26%) and 8,674,346 tail pixels (74%).The head was split again at a mean of 23.0.Source: §2 · p. 4
- The extraction procedure is applied to night-time imagery from 2010 to identify natural cities and examine their spatial interconnections.The available passage fragments mention 2010, natural cities, and interconnections, but provide no further quantitative details.Source: §2 · p. 4
- Natural cities are delineated from connected night-time-light pixels using a collective cutoff rather than individually imposed boundaries.The procedure is presented as a way to define cities consistently across the imagery.Source: §2 · p. 4
3. How t
The section illustrates Zipf’s law and related power-law plots with synthetic data, then describes the statistical tests used to assess city-size distributions. It favors maximum likelihood estimation and evaluates several city subsets without testing alternative distributions.
How Zipf’s law is represented and tested
- Ten synthetic city sizes following Zipf’s law produce straight lines in log-log plots of Zipf’s law, the Pareto CDF, and the power-law PDF.For the PDF, geometric bin widths are used; arithmetic progression can make the tail look messy or eliminate straight lines.Source: §3 · p. 6
- Least-squares estimation suffices for the illustration, but can produce unreliable or biased power-law results, so the study uses maximum-likelihood estimation.The authors describe maximum likelihood as the most rigorous statistical examination of power laws and use a p value to characterize goodness of fit.Source: §3 · p. 6
- For each city set, the study computes its size, power-law exponent, minimum city size, and goodness-of-fit p value, then tests all cities and two larger-city subsets.The subsets are cities above the minimum size and cities larger than average; alternative distributions are not examined.Source: §3 · p. 6
4. Results and discussion
Zipf’s law fits global natural-city sizes remarkably well across years, is weaker at the continental level, and is violated at the country level. City counts across countries also follow Zipf’s law.
4. Results and discussion
- The study compares about 30,000 natural cities in each of 1992, 2001, and 2010, using an average single-pixel cutoff of 31 across the three images.Zipf’s law was tested globally, continentally, and by country, for all cities, cities over 10 km², and cities larger than their average.Source: §4 · p. 6
- At the global level, Zipf’s exponent is exactly 1.0 and the law holds remarkably well across years; continentally it generally holds except in Africa in 2010, while country-level distributions violate it.Continental exponents range from 1.9 to 2.1, with p values greater than 0.02; the paper reports no country matching the global fit.Source: §4 · p. 6
- City counts among countries follow Zipf’s law: the largest country has twice as many natural cities as the second, three times as many as the third, with exponent 0.9–1.0.Country rankings also change over time: China moves from fourth to second, while Canada drops from third to sixth.Source: §4 · p. 8
5. Furth
The section discusses how geographic scope, time frame, and city delineation shape assessments of Zipf’s law, emphasizing naturally delineated cities and broad, dynamic views over administrative units and snapshots. It considers country-level departures in light of geographic and temporal limitations.
Further discussion
- Zipf’s law holds for cities considered as an interconnected whole, but may fail when the analysis isolates parts of that whole.The discussion contrasts a connected network of cities with studying individual countries or other subsets at particular times.Source: §5 · pp. 9–10
- Country-level violations can arise because country boundaries are not necessarily legitimate units for city-size distributions and snapshots omit broader dynamics.The discussion points to geographic units and static views as limitations on observing the whole system.Source: §5 · pp. 9–10
- Naturally delineated cities offer a better basis for studying city-size distributions than conventional administrative city definitions.The discussion emphasizes that conventional statistics and city definitions can diverge from naturally delineated urban areas.Source: §5 · p. 10
- Use remote-sensing or social-media data instead of conventional census data, which are aggregated, sampled, and small in size.The discussion contrasts these with data that are individual, all-encompassing, and big in size.Source: §5 · p. 11
- Use naturally delineated geographic units rather than top-down administrative boundaries, because human activities are not confined by country borders.The discussion attributes failures to observe Zipf’s law in some small countries mainly to country-boundary constraints, and notes globalization as a reason those constraints matter.Source: §5 · p. 11
- Assess Zipf’s law dynamically rather than at particular time instants, since cities continuously change and may grow toward an idealized status in which it appears.The discussion therefore urges caution when arguing against the law’s universality.Source: §5 · p. 11
6. Conclusion
Zipf’s law holds remarkably well for all natural cities globally, while country-level findings vary by place and time. The conclusion argues that observing the law requires appropriate data, scope, and methods.
6. Conclusion
- Zipf’s law holds remarkably well globally for natural cities, but is generally violated at country level despite exceptions by place and time.The authors argue that the law applies to cities as a whole rather than an arbitrary set such as one country.Source: §6 · p. 11
- The number of cities in the largest country is twice that in the second largest, three times that in the third largest, and so on.Source: §6 · p. 11
- The authors call for correct data, scope, and power-law methods, arguing conventional statistics, imposed geographic units, and least-squares estimates can yield questionable conclusions.Source: §6 · p. 11
- The three-year set of 30,000 natural cities per year could serve as benchmark data, and Table 3 indicates city expansion might be closely related to economic development.Source: §6 · p. 11
Appendix A: Power law detection for the 137 countries
The appendix reports power-law detection results for 137 of the 230 studied countries, with four parameters per country and guidance on interpreting fit and availability.
Appendix A: Power law detection for the 137 countries
- Each country’s results report city count, power-law exponent, minimum city size above which a power law appears, and goodness-of-fit index p.Source: Appendix A · p. 13
- A result with p set to N/A is not trustworthy; highlighted cells indicate Zipf’s law, while empty cells mean data are unavailable.Source: Appendix A · p. 13
- Afghanistan has fewer than six cities above the stated size threshold, too few for reliable statistical tests.The threshold is 10 square kilometers or mean, as stated in the appendix.Source: Appendix A · p. 13