Fractality and self-similarity in the structure of road networks

Fractal geometry has been an important tool in multiple disciplines, but it is usually limited to the geometrical fractal: a shape made of parts similar to the whole in certain ways. Recently there has been increasing interest in structural fractality of complex networks (e.g., biological and Internet hyperlinks) for studying organizational principles and evolutionary rules. In this study, the structural fractality of road networks (a kind of complex geographical network) is examined for better understanding of the complexity and dynamics of the road system. Fifty road networks of the most populous counties in the United States have been used as cases. The Maximum Excluded Mass Burning (MEMB) algorithm rooted in physics has been employed to compute the fractal dimensions by covering the network structure with a series of box sizes (ℓs). It is found that road networks have structural fractality. The values of structural fractal dimension range from 2.94 to 4.90 spanning ℓs= 5 to ℓs= 15. To examine the self-similar property of the structure of a road network at several scales, small-world and scale-free analyses have been carried out. It was found that road networks are structural self-similarities. The fractal and self-similar structures in a road network help improve the efficiency of flow transmission in the system and keep a balance between chaos and order. These findings help us understand the complexity, organizational rules, and dynamic principles of an urban system and blur the borders across several disciplines. They also provide an empirical guide for urban design and transportation planning.