Source-linked AI summary
Zipf's Law for All the Natural Cities in the United States: A Geospatial Perspective
Bin Jiang, Tao Jia
TL;DR
The paper addresses whether Zipf’s law holds for all U.S. cities despite subjective census boundaries. It constructs natural cities by clustering street nodes and finds that Zipf’s law holds remarkably well across the country, with little sensitivity to clustering resolution, unlike census urban areas.
Problem
Existing studies use census-defined city boundaries that are criticized as subjective or arbitrary, leaving the applicability of Zipf’s law to all U.S. settlements unresolved.
Method
The study clusters nationwide street nodes into natural cities without census information and compares them with census urban areas and population data.
Results
Zipf’s law holds remarkably well for all natural cities across the United States, with a Zipf value around 1.0 and little sensitivity to clustering resolution.
Takeaways & Limitations
The findings support a geospatial perspective in which natural-city definitions reveal stable power-law scaling across the entire United States.
Takeaways & Limitations
The analysis assumes that street-node patterns adequately reflect human settlements and that human activity requires street nodes.
Abstract
from arXiv · showhide
This paper provides a new geospatial perspective on whether or not Zipf's law holds for all cities or for the largest cities in the United States using a massive dataset and its computing. A major problem around this issue is how to define cities or city boundaries. Most of the investigations of Zipf's law rely on the demarcations of cities imposed by census data, e.g., metropolitan areas and census-designated places. These demarcations or definitions (of cities) are criticized for being subjective or even arbitrary. Alternative solutions to defining cities are suggested, but they still rely on census data for their definitions. In this paper we demarcate urban agglomerations by clustering street nodes (including intersections and ends), forming what we call natural cities. Based on the demarcation, we found that Zipf's law holds remarkably well for all the natural cities (over 2-4 million in total) across the United States. There is little sensitivity for the holding with respect to the clustering resolution used for demarcating the natural cities. This is a big contrast to urban areas, as defined in the census data, which do not hold stable for Zipf's law. Keywords: Natural cities, power law, data-intensive geospatial computing, scaling of geographic space
1. Introduction
The paper examines whether Zipf’s law applies across U.S. cities and addresses how subjective census boundaries affect that evaluation. It proposes defining all human settlements from street nodes rather than census classifications.
- Zipf’s law describes an inverse relationship between city size and rank, with a power-law exponent close to 1.0.
- Researchers debate whether Zipf’s law applies to all cities or primarily to the largest cities and upper tail.
- Census metropolitan areas and census-designated places impose legally or administratively determined boundaries that may exclude residents and appear arbitrary.
- The paper defines natural cities by clustering street nodes, including intersections and ends, without census information.
- The approach is intended to include settlements from an 8-million-resident megacity to a one-person town while reducing census-imposed bias.
2. Data
The study combines nationwide street-node data to construct natural cities with census urban-area and population data used for comparison. Natural cities are generated bottom up through recursive spatial clustering at a chosen resolution.
- The study uses about 25 million street nodes to derive 2-4 million natural cities and census data covering 3,638 urban areas and 65,997 population centers.
- Street nodes include intersections and ends, and the study treats them as indicators of human settlements because human activity is constrained to streets.
- The clustering algorithm recursively searches within a radius, adds nearby points, and continues until the connected point set is exhausted.
- Clustering resolution is the radius used by the algorithm; finer resolutions produce more natural cities, while 1-meter resolution reproduces the street-node count.
- Census urban areas were merged into 3,638 polygon units, while population centers record tract-level populations ranging from 1 to 36,146 people.
- Urban areas serve as reference data for comparison with the natural-city results and earlier studies using census-defined city units.
3. The mathematics of cities - Zipf’s law, power laws, and Pareto distributions
This section introduces power-law and Zipf-law concepts, methods for detecting power-law behavior, and procedures for estimating exponents and goodness of fit.
- Zipf’s law is a rank-size relation in which city size is inversely proportional to rank.
- A general power law is expressed as y = kx^-α, where k is a constant and α is the power-law exponent.
- Logarithmic plots can reveal power laws through straight-line patterns, but sparse tail bins introduce errors.
- The study adopts a modified KS test that compares empirical and fitted cumulative distributions using their maximum distance.
- The goodness-of-fit p-value is the fraction of 1,000 synthetic datasets whose KS distances exceed the observed distance, with values above 0.05 considered acceptable.
4. Results and discussion
Using street-node and physical-area measures, the study finds that Zipf’s law holds stably for natural cities across clustering resolutions, unlike census-defined urban areas.
- Natural-city sizes were measured using street nodes and physical areas, while census-defined urban areas were measured using population and physical areas.
- 4.1 The long tail of the distribution of natural cities: Around 90% of natural cities were smaller than the average size, while around 10% were larger, producing a pronounced long-tail imbalance.The authors describe this head–tail imbalance as an indicator of a power-law distribution.
- 4.2 Zipf’s Law for natural cities and urban areas: Zipf’s law held remarkably well for all natural cities at 400 m, 500 m, 600 m, and 700 m clustering resolutions.The corresponding power-law exponent was around 2.0, implying a Zipf exponent around 1.0.
- 4.2 Zipf’s Law for natural cities and urban areas: The natural-city power-law exponent changed only slightly across samples ranging from the biggest 150 cities to all cities.The reported changes were mostly confined to the second decimal place.
- 4.2 Zipf’s Law for natural cities and urban areas: Urban-area exponents were unstable across different parts of the tail and differed substantially from 2.0, indicating that they were not truly scale free.The study evaluated Zipf’s law with a modified KS test, whose reported values exceeded the 0.05 threshold.
- 4.2 Zipf’s Law for natural cities and urban areas: The study considered 500 m and 600 m the best clustering resolutions among the four options.The authors based this choice on visual comparison with urban areas and reasoning about the clustering algorithm; 1000 m would merge most real cities.
- 4.3 Zipf’s law for individual states: Within individual states, power laws held remarkably well for natural cities, but exponents varied across states and deviated from 2.0.The authors therefore state that Zipf’s law is not universal across geographical regions or countries in terms of its exponent.
5. Contributions of this paper
The paper’s contributions center on massive, bottom-up geospatial data and findings about Zipf’s law, while addressing sensitivity in alternative boundary methods and providing natural-city data for further research.
- 5. Contributions of this paper: Street-node boundary methods can be sensitive because kernel-density approaches depend on resolution, bandwidth, and density cutoffs.The adopted method involves only resolution when city size is measured by street-node count.
- 5. Contributions of this paper: Over 120 gigabytes of street-network data yielded 25 million street nodes and over dozens of millions of natural cities.Processing exhausted the available computing capacity and took days.
- 5. Contributions of this paper: The study’s novel aspects lie primarily in its massive data and geospatial findings, rather than in its boundary or power-law detection methods.The paper presents the scale and perspective as contributions to the long-standing debate about city-size distributions.
- 5. Contributions of this paper: The released natural-city dataset contains classified street nodes and is intended as a resource for geographic and broader scientific research.The resolution-600 data file is 1.2 GB and records ID, longitude, latitude, and class.
6. Conclusion
Using a geospatial perspective on all US human settlements, the study finds that Zipf’s law holds remarkably well nationwide, while state-level patterns and explanations for geographic scaling remain unresolved.
- 6. Conclusion: Zipf’s law holds remarkably well for all natural cities across the United States, with a Zipf value around 1.0.The conclusion concerns all human settlements or population concentrations included in the study.
- 6. Conclusion: Individual states do not show the same result, although urban areas remain power-law distributed with a Zipf value substantially different from 1.0.Urban areas are used as a proxy for city sizes in this comparison.
- 6. Conclusion: The study adds insights into geographic-space scaling by examining city distributions at a massive data level.The conclusion frames data-intensive computing as relevant to scientific discovery.
- 6. Conclusion: Street-segment connectivity is not power-law distributed, whereas street connectivity is, indicating that scaling depends on how geographic space is represented.The paper contrasts this property with conventional spatial heterogeneity characterized by a Gaussian distribution.
- 6. Conclusion: Future work will examine why different exponents occur across states or countries and how the observed regularity emerged.These questions remain open in the conclusion.