分形城市研究进展：从几何形态到网络关联

城市是人类文明最主要的聚集地,也是一个开放复杂演化巨系统,认识其空间复杂性成为城市科学研究的热点和前沿。城市研究正呈现“从位置和场所向关联和流,从物质空间向城市网络空间”的转向。人类对城市的认识也从“空间是机器”转向“城市有机生命体”。分形是大自然的语言和地理学第四代语言,是挖掘城市空间自组织规律的有效工具,在揭示城市有机体复杂演化机制方面发挥重要作用。目前分形城市研究以指标构建和实证分析为主,缺乏系统的梳理和回顾;内容上侧重几何分形,复杂城市网络视角下的结构分形研究薄弱。本文简要回顾了近30余年来分形城市研究的发展历程,梳理了分形城市研究脉络：从城市形态的几何分形到城市结构与关联的网络分形,从静态分形特征到动态异速生长,并展望了分形城市未来研究方向。

Advances in Fractal Cities: A Shift from Morphology to Network

The city is an accumulation of human civilization. It is also a highly complex system where a large number of agents interact, leading to a form and dynamics seemingly difficult to understand. Many studies in geography, ecology, sociology, economy and physics have been carried out to explore the general rules or regularities beneath the large number and the diverse agents operating in a city. It is widespread accepted that cities are an emergent phenomenon ruled by self-organization. As the language of nature and the fourth generation language of geography, fractal geometry has been a very powerful tool to capture the self-organized properties of cities. Most of the current studies are limited to the geometric fractal, i.e., based on fractal geometry: a shape made of parts similar to the whole in certain ways. Fractal geometry offers the significant advantages of capturing the spatial distribution, expansion, and filling properties of geographical objects in a city, and also describes the relationships between ranks and sizes of cities in an urban system. However, certain information—such as the efficiency of structural organization and the variance of levels of linkage—is ignored. As increasingly noticed by researchers, to better understand the ways a system of cities actually functions, we need to pay more attention to urban networks because current rapid developments of information and technology enable people to connect ever more easily and closely and in many new ways. This article reviews the advances of fractal cities from three aspects, which are geometric fractal, network fractal and evolutionary fractal. The significance and great potentials of fractal theory in urban studies are presented. The main research progress including fractal dimensions, fractal models, empirical studies and fractal cities and fractal urban systems are briefly reviewed, both for geometric and network fractals. As cities keep evolving, we also briefly review the evolutionary fractal cities, that is, the allometric scaling of cities. Based on current limitations on fractal cities, we propose a research agenda for fractal cities including (1) the development of measures and empirical studies on the third type of geographic fractals; (2) the spatial dependence and scale effects of fractal urban networks; (3) the coupling mechanics and influencing factors of fractal urban networks representing physical and non-physical urban spaces;(4) the DNA of a city from the perspective of fractals; and (5) the evolution simulation and policy intervention in fractal cities.