2016/10/31 by Yanguang Chen
Engineering · Environmental Science · Mathematics · Physics and Astronomy · #Artificial intelligence #Computer science #Entropy (arrow of time) #Fractal #Fractal analysis #Fractal dimension #Fractal dimension on networks #Fractal landscape #Geometry #Land Use and Ecosystem Services #Logistic function #Mathematical analysis #Mathematics #Physics #Scaling #Sigmoid function #Statistical physics #Statistics #Sustainability and Ecological Systems Analysis #Urban Design and Spatial Analysis #physics.soc-ph
paper · pdf · doi:10.5772/intechopen.68424
25 pages, 5 figures, 4 tables
arxiv created 2017/07/12 · openalex publication_date 2017/07/26 · arxiv updated 2017/08/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
A fractal is in essence a hierarchy with cascade structure, which can be described with a set of exponential functions. From these exponential functions, a set of power laws indicative of scaling can be derived. Hierarchy structure and spatial network proved to be associated with one another. This paper is devoted to exploring the theory of fractal analysis of complex systems by means of hierarchical scaling. Two research methods are utilized to make this study, including logic analysis method and empirical analysis method. The main results are as follows. First, a fractal system such as Cantor set is described from the hierarchical angle of view; based on hierarchical structure, three approaches are proposed to estimate fractal dimension. Second, the hierarchical scaling can be generalized to describe multifractals, fractal complementary sets, and self‐similar curve such as logarithmic spiral. Third, complex systems such as urban systems are demonstrated to be a self-similar hierarchy. The human settlements in Germany and the population of different languages in the world are taken as two examples to conduct empirical analyses. This study may reveal the association of fractal analysis with other types of scaling analysis of complex systems, and spatial optimization theory may be developed in future by combining the theories of fractals, allometry, and hierarchy.