It is widely recognized that geographic features such as mountains and rivers are neither smooth nor regular, so they cannot be precisely described on the basis of Euclidean geometry. Instead, fractal geometry has the capacity to fill this gap. The very word “fractal” originates from Latin and literally means “broken” or “irregular”. The development of fractal geometry went through three definitions, all of which center on the term “self-similarity”: a part of an object or pattern is similar to the whole [1
]. For the first and second definitions, fractals refer to shapes that are either strictly or statistically self-similar. A classic example of strictly self-similar shapes is a Koch curve [4
], whose parts are exactly the scaled copies of the original curve. Mandelbrot [5
] relaxed the strictness by adding some randomness to those “scaled copies” to make the constructed curve capable of describing real-world features, such as a coastline. Strict and statistical self-similarities both maintain the same power law exponent or the fractal dimension from the power law relationship between the number of copies in the fractal and the scale at which it is measured. As the power law relationship is too tough to universally see the real-world fractals, Jiang and Yin [6
] further relaxed this statistical requirement by checking if the pattern of scaling—far more small scales than large ones—of a fractal recurs at least twice. In the third definition, the focus of self-similarity is no longer on the geometric form, but rather on the recurring pattern of scaling statistics among iteratively derived sub-wholes of data.
Urban space exhibits fractal patterns from two main perspectives. Morphologically, a large body of literature presents the fractal evidence from many urban elements—for instance, built-up areas, land uses, streets, and boundary shapes [7
], and their evolution or growth (e.g., [13
]). Functionally, human activities inside an urban space, especially those manifested by location-based social media data, agglomerate into a fractal-like structure that contains a minority of high-density locations and a majority of low-density ones, which are ordered spatially from the city center to the periphery [16
]. Such spatial configuration of human activities is also claimed to be akin to the fractal structure of central place systems [18
]. Among those aforementioned city elements, urban streets are most broadly adopted for understanding both a city’s form and function. On one hand, streets represent a city’s physical environment and showcase the coherency of its spatial organization. On the other hand, human urban activities or movements are, to a large degree, confined within the network of streets. Therefore, it is important to examine fractal cities from the perspective of street networks in order to obtain new insights into a city’s underlying spatial environment, its configuration, and the human activities therein.
As one particular type of complex networks whose nodes and links are located in the geographic space, street networks are naturally modeled as graph representations, wherein the dual graph (nodes as streets, links as intersections; [19
]) enable us to analyze the street structural configuration through a set of topological measures based on graph theory [20
] and space syntax approaches [22
]. Similar to many other real-world complex networks, such as social networks and protein interactions, the dual graph of street networks tend to possess a fractal or scaling property with respect to the power law distribution of the node degrees [20
], or the power law relationship between the number of non-overlapping compartments covering the entire network and their sizes [24
]. However, because the power law model is too strict to measure a fractal whose statistical distribution follows a non-power-law but still appears to have fat-tailed characteristics (e.g., log-normal), the assessment of whether street networks are fractals across a large variety of cities is challenging. Then, this paper is motivated to use the third definition of fractal, which unties the power law constraint by considering other types of heavy-tailed distribution as the support of fractality, to identify the universal fractal pattern of street networks at a wide spatial extent.
Another triggering factor of this research is the geospatial big data. The limited data availability and computational resources in the past has led to a relatively narrow spatial scope of works belonging to urban street networks; that is, a neighborhood or city level [27
]. Big data offers unprecedented opportunities for researchers to achieve the massive-scale geospatial data sets. For example, through OpenStreetMap (OSM), a volunteered geospatial information platform, we are able to achieve and process the street network nationwide, or even worldwide coverage, which creates new opportunities to investigate street fractalities at intercity levels within a country or across countries [28
]. Not only can the national road network be accessed, but other national urban quantities can be easily accessed as well, such as Gross Domestic Product (GDP) and CO2
emissions. This opens the possibility for us to unpack the relationship between street-network configuration and urban energy, environmental, and economic indicators. Prior urban-scale studies have reported that streets with fractal features can contribute to the enhancement of social capital and the reduction of greenhouse gas emissions because of larger street-network efficiency (e.g., [26
]). This study would conduct such exploratory analysis on the country scale.
The contribution of this study is three-fold. Firstly, we employed the natural street representation (a continuous street rather than its contained meaningless segments or vertices; see the details in Section 2.2
) for the identification of the fractal or scaling structure. We have successfully processed and derived the natural streets for almost all prefectural-level cities in China, which could be a valuable data source for the research community. Second, we designed an innovative methodology in combining different fractal metrics under the theoretical framework of the third definition. Starting from the head/tail breaks-induced ht-index [6
], we can assess the fractality of the street network for each individual city and the overall complexity of all street networks at a country scale. Next, due to the lack of sensitivity of the ht-index, we employed two alternatives: the cumulative rate of growth (CRG, [30
]) and the ratio of areas in a rank-size plot (RA, [31
]) (see more details in Section 2.3
). We found that using them collectively can differentiate cities within the same scaling hierarchy, making it possible to formulate a more comprehensive and systematic understanding of Chinese urban forms. Lastly, we sought relationships between the fractality of street networks and different kinds of urban metrics and evaluated the explanatory power of urban fractals on urban socioeconomic status and energy consumption at the national scale.
The remainder of this paper proceeds as follows. Section 2
introduces the data sets and the proposed methodological framework. Section 3
presents the visualization and statistical results regarding the universal fractal pattern over nearly 300 cities in China, as well as the correlations between each fractal metric and urban quantity. Section 4
further discusses the fractal structure of street networks, before Section 5
draws conclusions and points to future research directions.
4. Further Discussion of this Study
The augmented openness and attainability of geospatial big data helps us to analyze and grasp urban forms at both intracity and intercity levels. Street networks largely reflect an overall picture of an urban physical environment. Based on the derived natural streets over 298 cities, we can identify the fractal structure and scaling statistics of China urban street networks and further find the universal fractal pattern among a big variety of urban spaces. In this section, we further discuss the developed methodology and the obtained results and how they contribute to our knowledge in current Chinese urban studies.
Geographic features are fractal in essence, as is the urban space that involves a large collection of geographic features. The invention of the third definition of fractal [6
] enables us to perceive the universal fractal pattern at various scales of geographic space. It should be stressed that given the right perspective, almost all geographic features can be seen as fractals. For example, previous work explores the underlying fractality from an individual smooth curve [1
] from the very perspective of recursive bends rather than its contained vertices or segments. Similarly, for the collection of linear features—that is, the street network—one must take the perspective of street–street topology instead of segment–segment or junction–junction topology for seeing the fractal. In this study, we employed such perspective and demonstrated successfully the universal fractal pattern of street networks among over 298 cities.
The detected universal fractal pattern of urban street networks can be captured quantitatively by the power-law distribution of street connectivities and the underlying scaling hierarchies. Power-law distribution is widely observed in natural and societal phenomena, and it is usually regarded as an indicator for sustainability [29
]. The power-law detection for each city was conducted using the robust maximum likelihood estimation method and helps us identify that about 67% of Chinese urban street networks hold a power law nature. For the remaining 33% of cities, the scaling hierarchical levels derived from head/tail breaks show, as well, the striking fractal patterns of far more less-connected streets than well-connected ones. Thus, the ht-index is superior to power-law metrics with respect to the identification of the universal fractal pattern of Chinese street networks. Besides, the ht-index is good for discriminating urban street networks across different scaling hierarchies; its two alternatives, CRG and RA, help us to differentiate the cities within the same scaling hierarchy because of the increment of sensitivity. In this regard, the integration of three fractal measures can provide us with a more comprehensive and also a more precise picture of fractal cities in China.
The study also attempted to estimate the explanatory power of the fractality of street networks on urban socioeconomic status and fuel consumption. As the urban indicators such as GDP are normally associated with multiple factors of a street network, the study adopted the multiple regression approach by involving other traditional street-network geometric measures in order to make the model closer to the reality and reduce the probability of confounding effects. As a result, the correlation coefficient R2 value of the multiple regression model, as in the case with GDP (R2 = 0.57), is higher than that of a single regression model based on one single street geometric property or fractal metric (e.g., R2 = 0.41 with only or R2 = 0.23 with CRG only). Therefore, it can be said that multiple street-network measures are suitable for an exploratory analysis of each urban quantity than that of a single measure.
Our multiple regression analysis further suggests that cities with more strikingly fractal street patterns (i.e., more scaling hierarchical levels of their street networks) tend to have better economic status. The fractal analysis results in Table 4
also confirm that those most fractal cities are top-tier metropolitans in China such as Beijing and Shanghai. If we take a closer look at the coefficient
, it should be noted that the positive effect of
to urban socioeconomic status is stronger than any of three fractal metrics. This indicates that a city’s economic development is more directly represented by the geometric details of its street network than by the underlying fractal pattern. However, it does not mean that the fractal metrics are less meaningful. As shown by the regression analysis results with CO2
emissions, the fractal structure of a street network can contribute to the energy efficiency of a city. The opposite signs of
between fractal metrics and
indicate that street fractalities can perform differently from street geometric properties. In this regard, the fractality of street networks can be a useful supplement to traditional street-network configuration measures.