Zipf's Law for All the Natural Cities around the World

Abstract: 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

• The paper confirms that Zipf’s law robustly applies to natural cities worldwide with a consistent exponent close to 1.0.
• The study reveals continental and country-level variations, noting exceptions such as in Africa due to smaller samples and temporal changes.
• The methodology employs night-time satellite imagery to objectively delineate cities, challenging conventional census-based approaches.

Evaluation of Zipf’s Law Across Global Urban Settings

Bin Jiang, Junjun Yin, and Qingling Liu's paper titled "Zipf’s Law for All the Natural Cities Around the World" offers a comprehensive examination of Zipf's law within the context of global urban environments. This study stands apart from others by testing the universality of Zipf’s law using a unique delineation of cities identified through night-time satellite imagery, thereby offering novel insights into the scalability and applicability of this law across different geographic and temporal contexts.

Core Findings and Methodology

The cornerstone of the paper is the verification of Zipf's law globally, embracing the concept of naturally delineated cities. These "natural cities" are derived from night-time imagery, which provides an objective and consistent method for city identification across the globe. The researchers utilize three pivotal time points (1992, 2001, and 2010) to evaluate the stability and persistence of Zipf's law—a power-law distribution where the logarithm of city rank is proportionally inverse to city size.

The paper offers the following key findings:

1. Global Organization: Zipf’s law holds robustly for all natural cities at a global level, with a consistent exponent (α ≈ 1.0) observed across the entire dataset. This consistency underscores the potential universality of Zipf’s law if applied at the correct scope.
2. Continental Variations: While generally valid at the continental scale, notable exceptions, such as Africa during specific time periods, suggest fluctuations in compliance, attributable to both temporal changes and the smaller sample size in some regions.
3. Country-Specific Fluctuations: At a country level, findings are less uniform, with notable violations of Zipf’s law. However, these discrepancies are framed in the context of sample sizes and temporal snapshots, indicating that such deviations do not negate the law's overall applicability.
4. City Number Distribution: The study extends Zipf’s law beyond city size distribution, leading to the assertion that the number of cities per country adheres to a similar proportional pattern: the number of cities in a primary country is twice that in the second, and so on.

Implications and Future Research Directions

The implications of this study are both methodological and conceptual. Methodologically, the adoption of night-time imagery to delineate natural cities circumvents some of the constraints of conventional census-based approaches, addressing criticisms about biases induced by administrative boundaries. The approach enriches the urban analysis toolkit, providing a data-rich layer that supports more nuanced urban studies.

Conceptually, the assertion of Zipf's law’s universality invites a reevaluation of traditional urban growth theories. It challenges researchers to consider cities as interconnected entities beyond national borders, promoting a global network perspective in urban studies. This shift could redefine approaches to urban planning and socioeconomic modeling, encouraging integrated frameworks that transcend local scales.

Future research could build upon these findings by exploring the mechanisms that sustain the Zipfian distribution at multiple scales. This could involve integrating socio-economic data with natural city delineations to untangle the interdependencies influencing urban growth. Additionally, longitudinal studies that extend beyond the three-decade scope of this work could shed light on the temporal evolution of these distributions.

In conclusion, the paper by Jiang et al. contributes a substantial dataset and a robust framework for examining Zipf’s law in the context of global urban systems. Its reliance on big data and innovative methodologies positions it as a significant reference point for further research into urban dynamics and power law distributions.