Urban Allometric Scaling Beneath Structural Fractality of Road Networks

Abstract

Allometry originally referred to the scaling relationship between the size of a body part and the size of the whole body when an organism grows. Gradually, researchers introduced it into urban studies. In existing urban studies, many allometric relations were discovered, especially those between urban population and other (physical or socioeconomic) quantities, such as urban area and gross domestic product (GDP). Recently, geometric fractal dimension ($D_g$) was used as a complexity measure of road networks and a linear relationship between $D_g$ and urban population was found. The complexity of a road network is related not only to its geometric form, which can be described by $D_g$, but also to its topological structure, which can be described by structural fractal dimension ($D_s$). Whether some relations, such as allometric relations, exist between $D_s$ and other urban quantities is vague. This study explores the allometric relations between the $D_s$ of urban road networks and urban quantities in Hong Kong for the period from 1971 to 2011. It is found that $D_s$ has positive allometric relations with population, CO₂ emissions, GDP, merchandise imports, and merchandise exports (with scaling exponents of 1.581, 4.298, 11.113, 13.951, and 14.141, respectively) but inverse allometric relations with the areas of arable and agricultural land (with scaling exponents of −2.857 and −1.918, respectively). These findings indicate that $D_s$ not only has allometric relations with urban quantities but also could have different types of relations. These discoveries could form another basis for the study of urban development.

异速增长原本指涉器官成长时，身体某部分的大小和整体大小之间的比率关系，研究者则逐渐将该概念引入城市研究之中。既有的城市研究发现诸多异速增长关系，特别是在城市人口与其他（实体或社会经济）数量之间，诸如城市面积与国内生产总值（GDP）。几何碎形维度（Dg）晚近被用来作为路网复杂度的测量方法，并发现Dg和城市人口之间存在线性关系。路网的复杂度，并非仅与可由Dg描绘的几何形式有关，而同时与其通过结构碎形维度（Ds）描绘的拓朴结构有关。诸如异速增长关系等若干关系是否存在于Ds和其他城市数量之间，则是模糊的。本研究探讨香港在1971年至2011年间，城市路网的Ds和城市数量之间的异速增长关系。本研究发现，Ds与人口、二氧化碳排放、GDP、商品进口和出口之间具有正向的异速增长关系（比率指数分别为1.581、4.298、11.113、13.951与14.141），但与可耕地面积和农业用地之间则存在着反向的异速增长关系（比率指数分别为2.857和1.918）。这些发现意味着Ds不仅和城市数量间存在异速增长关系，同时可能有不同的关系类型。这些发现能够为城市发展研究形成另一个基础。

La alometría originalmente se refería a la relación de escala entre el tamaño de una parte del cuerpo y el tamaño de todo el cuerpo cuando un organismo crece. Gradualmente, los investigadores introdujeron el término en los estudios urbanos. En los actuales estudios urbanos se descubrieron muchas relaciones alométricas, especialmente las que ocurren entre la población urbana y otras cantidades (físicas o socioeconómicas), tales como área urbana y producto nacional bruto (PNB). Recientemente, se usó la dimensión fractal geométrica (Dg) como una medida de la complejidad de las redes de carreteras, y se encontró una relación lineal entre Dg y la población urbana. La complejidad de una red de carreteras está relacionada no solo con su forma geométrica, que puede describirse con Dg, sino también con su estructura topológica, que puede describirse con la dimensión fractal estructural (Ds). El que existan algunas relaciones, tales como las relaciones alométricas, entre Ds y otras cantidades urbanas, es algo vago. Este estudio explora las relaciones alométricas entre la Ds de las redes viales urbanas y cantidades urbanas en Hong Kong, durante el período 1971–2011. Se encuentra que Ds tiene relaciones alométricas positivas con población, emisiones de CO2, PNB, importación de mercancías y exportaciones de mercancía (con exponentes de escala de 1.581, 4.298, 11.113, 13.951 y 14.141, respectivamente) pero relaciones alométricas inversas con las áreas de tierra arable y tierra agrícola (con exponente de escala de –2.857 y –1.918, respectivamente). Estos hallazgos indican que Ds no solo tiene relaciones alométricas con cantidades urbanas sino también podría tener diferentes tipos de relaciones. Estos descubrimientos podrían constituir otra base para el estudio del desarrollo urbano.

Key Words:

• road networks
• structural fractality
• urban allometric scaling.

关键词:

• 路网
• 结构碎形
• 城市异速增长比率。

Palabras clave:

• escala alométrica urbana
• fractalidad estructural
• redes de carreteras.

We thank the editor and the anonymous reviewer for their valuable comments, which led to significant improvement in the quality of this article. We also thank Wenjia Wu at Tsinghua University and Xintao Liu at the Hong Kong Polytechnic University for their helpful discussions and suggestions.

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Additional information

Funding

This work was supported by the National Natural Science Foundation of China (41471383) and the Hong Kong Research Grant Council (PolyU 152233/15E).

Notes on contributors

Tian Lan

TIAN LAN was a graduate student in the Faculty of Geosciences and Environmental Engineering at Southwest Jiaotong University when the early work of this study began. Now he is a PhD Candidate in the Department of Land Surveying and Geo-Informatics at the Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong. E-mail: [email protected]. His research interests include automated cartography and fractal and complex network analysis.

Zhilin Li

ZHILIN LI is the Chair Professor of Geo-Informatics in the Department of Land Surveying and Geo-Informatics at the Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong. E-mail: [email protected]. His research interests include digital cartography, scale-driven spatial data modeling and analysis, and feature extraction from remote sensing images.

Hong Zhang

HONG ZHANG is an Associate Professor in the Faculty of Geosciences and Environmental Engineering at Southwest Jiaotong University, Chengdu, China. E-mail: [email protected]. Her research interests include urban geography and multiscale modeling.