Cities are spatially evolving complex systems. The order and pattern beneath the apparent chaos and diverse physical forms of cities are still unclear. How the form and structure of a city evolve to improve its functions needs further exploration. To fill thisgap, we examine the geometric fractal (GF), the topological fractal (TF), and the hierarchical fractal (HF) evolution of cities by taking Hong Kong street networks from year 1971 to 2018 as an example. We find that these networks keep to be fractals both in form and structure. The values of GF, TF, and HF dimensions increase with fluctuations, revealing a more mature and complex street network. The radius-length GF dimensions demonstrate the bi-fractal property, with values ranged 1.653–1.832 and 0.677–0.892, respectively, reflecting a core-periphery pattern. The values of TF dimensions increase steadily with a wider gap to GF dimensions, indicating progressively structural optimization of street networks. These street networks keep showing fractal properties in form and structure through spatial extension, local densification, vertical stratification, hierarchies enrichment, and shortcuts construction. Moreover, street networks are GFs and TFs at the city, county, and MSA scales. The discoveries advance our understanding of urban development.

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.

Schematic maps are popular for the representation of transport networks. Many automated methods have been developed to generate such maps. In these methods, optimization techniques work with various sets of constraints. Most of these constraints govern geometric properties of individual features. A few constraints address relationships among features, but none explicitly deal with the main structure of an entire network. We believe that preservation of main structure is the most important and preservation of relative relations is helpful. This is because human perception follows a global-to-local process. These constraints have recently been formed into four general principles, with two for global structure and two for relativity of features. This study develops an automated method by integrating these principles into the mixed-integer programming (MIP) framework. Experimental evaluations have been conducted with two sets of real-world transit networks. In comparison to the existing method, the proposed method has smaller fractal dimensions, better computational performance and higher scores in terms of clarity, recognition of major lines, visual simplicity and satisfaction. Therefore, it is concluded that the proposed method can generate schematic maps with improved clarity and aesthetics. The idea in this study is also helpful for the design of other visual representations.

Organisms and other living things grow as fractals, helping to maximize space-filling ability and transmit flow efficiently. In recent years, many studies on fractals have been carried out, mostly at a geometric level, although little work has been done on fractal properties at a structural level. In fact, the capacity of geographical objects, especially road networks, is dependent not only on the geometric distribution described by a geometric fractal dimension but also on the level of structural complexity as described by a structural fractal dimension. Past studies on a structural fractal dimension are limited to a single scale and a single representation. Because different representations could lead to different results in fractal analysis, this study aims to examine the structural fractality of urban road networks by different representations. The road networks of the 100 most populous cities in the United States were examined by three commonly used topological representations. The following results were established: Despite differences in representation, structural fractality always exists; structural fractal dimensions in stroke-based representation are obviously larger than those in segment-based representation; and scale has no obvious effects on structural fractality. These results could deepen our understanding of how urban road networks evolve.