Fractal structure of a system suggests the optimal way in which parts arranged or put together to form a whole. The ideas from fractals have a potential application to the researches on urban sustainable development. To characterize fractal cities, we need the measure of fractional dimension. However, if the fractal organization is concealed in the complex spatial distributions of geographical phenomena, the common methods of evaluating fractal parameter will be disabled. In this article, a new model is proposed to describe urban density and estimate fractal dimension of urban form. If urban density takes on quasi-fractal pattern or the self-similar pattern is hidden in the negative exponential distribution, the generalized gamma function may be employed to model the urban landscape and estimate its latent fractal dimension. As a case study, the method is applied to the city of Hangzhou, China. The results show that urban form evolves from simple to complex structure with time.
The conventional mathematical methods are based on characteristic length, while urban form has no characteristic length in many aspects. Urban area is a measure of scale dependence, which indicates the scale-free distribution of urban patterns. Thus, the urban description based on characteristic lengths should be replaced by urban characterization based on scaling. Fractal geometry is one powerful tool for scaling analysis of cities. Fractal parameters can be defined by entropy and correlation functions. However, how to understand city fractals is still a pending question. By means of logic de…
The fractal dimension growth of urban form can be described with sigmoid functions such as logistic function due to squashing effect. The sigmoid curves of fractal dimension suggest a type of spatial replacement dynamics of urban evolution. How to understand the underlying rationale of the fractal dimension curves is a pending problem. This study is based on two previous findings. First, normalized fractal dimension proved to equal normalized spatial entropy; second, a sigmoid function proceeds from an urban-rural interaction model. Defining urban space-filling measurement by spatial entropy, …
Urban form has been empirically demonstrated to be of scaling invariance and can be described with fractal geometry. However, the rational range of fractal dimension value and the relationships between various fractal indicators of cities are not yet revealed in theory. By mathematical deduction and transformation (e.g. Fourier transform), I find that scaling analysis, spectral analysis, and spatial correlation analysis are all associated with fractal concepts and can be integrated into a new approach to fractal analysis of cities. This method can be termed 3S analyses of urban form. Using the…
Urban population density always follows the exponential distribution and can be described with Clark's model. Because of this, the spatial distribution of urban population used to be regarded as non-fractal pattern. However, Clark's model differs from the exponential function in mathematics because that urban population is distributed on the fractal support of landform and land-use form. By using mathematical transform and empirical evidence, we argue that there are self-affine scaling relations and local power laws behind the exponential distribution of urban density. The scale parameter of C…
The time series of fractal dimension values of urban form always take on sigmoid curves. The basic model of these curves is logistic function. From the logistic model of fractal dimension curves, we can derive the growth rate formula and acceleration formula of city development. Using the inflexions of the fractal parameter curves, we can identify the different phases of urban evolution. The main results are as follows. (1) Based on the curve of fractal dimension of urban form, urban growth can be divided into four stages: initial slow growth, accelerated fast growth, decelerated fast growth, …
The improved city clustering algorithm can be used to identify urban boundaries on a digital map, and the results are a set of isolines. The relationships between the urban measurements within the variable boundaries follow allometric scaling law, which indicates spatial allometry of cities. This paper is devoted to exploring the fractal dimension proceeding from urban spatial allometry. By theoretical reasoning and empirical analysis of urban traffic network, we can derive a set of fractal dimension from the spatial allometry and reveal the basic property of the new fractal parameters. The fi…
Fractal dimension curves of urban growth can be modeled with sigmoid functions, including logistic function and quadratic logistic function. Different types of logistic functions indicate different spatial dynamics. The fractal dimension curves of urban growth in western countries follows the common logistic function, while these curves of cities in northern China follows quadratic logistic function. Now we want to know whether all Chinese cities follow the same rules of urban evolution. This paper is devoted to exploring the fractals and fractal dimension properties of the city of Shenzhen in…
Urban form and growth can be described with fractal dimension, which is a measurement of space filling of urban evolution. Based on empirical analyses, a discovery is made that the time series of fractal dimension of urban form can be treated as a sigmoid function of time. Among various sigmoid functions, the logistic function is the most probable selection. However, how to use the model of fractal dimension growth to explain and predict urban growth is a pending problem remaining to be solved. This paper is devoted to modeling fractal dimension evolution of different types of cities. A intere…
A type of fractal dimension definition is based on the generalized entropy function. Both entropy and fractal dimension can be employed to characterize complex spatial systems such as cities and regions. Despite the inherent connect between entropy and fractal dimension, they have different application scopes and directions in urban studies. This paper focuses on exploring how to convert entropy measurement into fractal dimension for the spatial analysis of scale-free urban phenomena using ideas from scaling. Urban systems proved to be random prefractal and multifractals systems. The entropy o…
Spatial patterns and processes of cities can be described with various entropy functions. However, spatial entropy always depends on the scale of measurement, and it is difficult to find a characteristic value for it. In contrast, fractal parameters can be employed to characterize scale-free phenomena. This paper is devoted to exploring the similarities and differences between spatial entropy and fractal dimension in urban description. Drawing an analogy between cities and growing fractals, we illustrate the definitions of fractal dimension based on different entropy concepts. Three representa…
Entropy is one of physical bases for fractal dimension definition, and the generalized fractal dimension was defined by Renyi entropy. Using fractal dimension, we can describe urban growth and form and characterize spatial complexity. A number of fractal models and measurements have been proposed for urban studies. However, the precondition for fractal dimension application is to find scaling relations in cities. In absence of scaling property, we can make use of entropy function and measurements. This paper is devoted to researching how to describe urban growth by using spatial entropy. By an…
This paper presents a new perspective of looking at the relation between fractals and chaos by means of cities. Especially, a principle of space filling and spatial replacement is proposed to explain the fractal dimension of urban form. The fractal dimension evolution of urban growth can be empirically modeled with Boltzmann's equation. For the normalized data, Boltzmann's equation is equivalent to the logistic function. The logistic equation can be transformed into the well-known 1-dimensional logistic map, which is based on a 2-dimensional map suggesting spatial replacement dynamics of city …
Fractal geometry provides a powerful tool for scale-free spatial analysis of cities, but the fractal dimension calculation results always depend on methods and scopes of study area. This phenomenon has been puzzling many researchers. This paper is devoted to discussing the problem of uncertainty of fractal dimension estimation and the potential solutions to it. Using regular fractals as archetypes, we can reveal the causes and effects of the diversity of fractal dimension estimation results by analogy. The main factors influencing fractal dimension values of cities include prefractal structure…
The area-perimeter allometric scaling is a basic and important approach for researching fractal cities and has been studied for a long time. However, the boundary dimension of a city is always numerically overestimated by the traditional formula. An adjusting formula has been derived to revise the overestimated boundary dimension and estimate the form dimension, but the association between the global and local fractal parameters is not clear. This paper is devoted to describing the urban evolution by using the improved fractal parameters based on the area-perimeter measure relation. A system o…
The area-perimeter scaling can be employed to evaluate the fractal dimension of urban boundaries. However, the formula in common use seems to be not correct. By means of mathematical method, a new formula of calculating the boundary dimension of cities is derived from the idea of box-counting measurement and the principle of dimensional consistency in this paper. Thus, several practical results are obtained as follows. First, I derive the hyperbolic relation between the boundary dimension and form dimension of cities. Using the relation, we can estimate the form dimension through the boundary …
For a long time, many methods are developed to make temporal signal analyses based on time series. However, for geographical systems, spatial signal analyses are as important as temporal signal analyses. Nonstationary spatial and temporal processes are associated with nonlinearity, and cannot be effectively analyzed by conventional analytical approaches. Fractal theory provides a powerful tool for exploring complexity and is helpful for spatio-temporal signal analysis. This paper is devoted to researching spatial signals of geographical systems by means of wave-spectrum scaling. The traffic ne…
The growth curves of fractal dimension of urban form take on squashing effect and can be described by sigmoid functions. The fractal dimension growth of urban form in western countries can be modeled by Boltzmann's equation and logistic function. However, these models cannot be well applied to the fractal dimension growth curve of Beijing city, the national capital of China. In this paper, the experimental method is employed to find parametric models for the growth curves of fractal dimension of Chinese urban form. By statistical analysis, numerical analysis, and comparative analysis, we find …
The gravity model is one of important models of social physics and human geography, but several basic theoretical and methodological problems remain to be solved. In particular, it is hard to explain and evaluate the distance exponent using the ideas from Euclidean geometry. This paper is devoted to exploring the distance-decay parameter of the urban gravity model. Based on the concepts from fractal geometry, several fractal parameter relations can be derived from scaling laws of self-similar hierarchies of cities. Results show that the distance exponent is just a scaling exponent, which equal…
This paper proposes a new method to analyze the spatial structure of urban systems using ideas from fractals. Regarding a system of cities as a set of "particles" distributed randomly on a triangular lattice, we construct a spatial correlation function of cities. Suppose that the spatial correlation follows the power law. It can be proved that the correlation exponent is the second order generalized dimension. The spatial correlation model is applied to the system of cities in China. The results show that the Chinese urban system can be described by the correlation dimension ranging from 1.3 t…
An analogy between the fractal nature of networks of arteries and that of systems of rivers has been drawn in the previous works. However, the deep structure of the hierarchy of blood vessels has not yet been revealed. This paper is devoted to researching the fractals, allometric scaling, and hierarchy of blood vessels. By analogy with Horton-Strahler's laws of river composition, three exponential laws have been put forward. These exponential laws can be reconstructed and transformed into three linear scaling laws, which can be named composition laws of blood vessels network. From these linear…
Traffic networks have been proved to be fractal systems. However, previous studies mainly focused on monofractal networks, while complex systems are of multifractal structure. This paper is devoted to exploring the general regularities of multifractal scaling processes in the street network of 12 Chinese cities. The city clustering algorithm is employed to identify urban boundaries for defining comparable study areas; box-counting method and the direct determination method are utilized to extract spatial data; the least squares calculation is employed to estimate the global and local multifrac…
The boundaries of central place models proved to be fractal lines, which compose fractal texture of central place networks. A textural fractal can be employed to explain the scale-free property of regional boundaries such as border lines, but it cannot be directly applied to spatial structure of real human settlement systems. To solve this problem, this paper is devoted to deriving structural fractals of central place models from the textural fractals. The method is theoretical deduction based on the dimension rules of fractal sets. The textural fractals of central place models are reconstruct…
Just as natural river networks are known to be globally self-similar, recent research has shown that human-built urban networks, such as road networks, are also functionally self-similar, and have fractal topology with power-law node-degree distributions (p(k) = a k). Here we show, for the first time, that other urban infrastructure networks (sanitary and storm-water sewers), which sustain flows of critical services for urban citizens, also show scale-free functional topologies. For roads and drainage networks, we compared functional topological metrics, derived from high-resolution data (70,0…
Urban scaling laws relate socio-economic, behavioral, and physical variables to the population size of cities and allow for a new paradigm of city planning, and an understanding of urban resilience and economies. Independently of culture and climate, almost all cities exhibit two fundamental scaling exponents, one sub-linear and one super-linear that are related. Here we show that based on fundamental fractal geometric relations of cities we derive both exponents and their relation. Sub-linear scaling arises as the ratio of the fractal dimensions of the road network and the distribution of the…
The spatial distributions of cities fall into two groups: one is the simple distribution with characteristic scale (e.g. exponential distribution), and the other is the complex distribution without characteristic scale (e.g. power-law distribution). The latter belongs to scale-free distributions, which can be modeled with fractal geometry. However, fractal dimension is not suitable for the former distribution. In contrast, spatial entropy can be used to measure any types of urban distributions. This paper is devoted to generalizing multifractal parameters by means of dual relation between Eucl…
Central place systems have been demonstrated to possess self-similar patterns in both the theoretical and empirical perspectives. A central place fractal can be treated as a monofractal with a single scaling process. However, in the real world, a system of human settlements is a complex network with multi-scaling processes. The simple fractal central place models are not enough to interpret the spatial patterns and evolutive processes of urban systems. It is necessary to construct multi-scaling fractal models of urban places. Based on the postulate of intermittent space filling, two typical mu…
Highly accurate estimates of the urban fractal dimension $D_f$ are obtained by implementing the Detrended Moving Average algorithm (DMA) on WorldView2 satellite high-resolution multi-spectral images covering the largest European cities. Higher fractal dimensions are systematically obtained for urban sectors (centrally located areas) than for suburban and peripheral areas, with $Df$ values ranging from $1.65$ to $1.90$ respectively.The exponents $β_s$ and $β_i$ of the scaling law $N^β$ with $N$ the population size, respectively for socio-economic and infrastructural variables, are evaluated for…
Fractal dimension is defined on the base of entropy, including macro state entropy and information entropy. The generalized correlation dimension of multifractals is based on Renyi entropy. However, the mathematical transform from entropy to fractal dimension is not yet clear in both theory and practice. This paper is devoted to revealing the new equivalence relation between spatial entropy and fractal dimension using functional box-counting method. Based on varied regular fractals, the numerical relationship between spatial entropy and fractal dimension is examined. The results show that the …
Complex systems have become a popular lens for analyzing cities and complexity theory has many implications for urban performance and resilience. This paper develops a typology of measures and indicators for assessing the physical complexity of the built environment at the scale of urban design. It extends quantitative measures from city planning, network science, ecosystems studies, fractal geometry, statistical physics, and information theory to the analysis of urban form and qualitative human experience. Metrics at multiple scales are scattered throughout diverse bodies of literature and ha…
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 Canto…
The central place models are fundamentally important in theoretical geography and city planning theory. The texture and structure of central place networks have been demonstrated to be self-similar in both theoretical and empirical studies. However, the underlying rationale of central place fractals in the real world has not yet been revealed so far. This paper is devoted to illustrating the mechanisms by which the fractal patterns can be generated from central place systems. The structural dimension of the traditional central place models is d=2 indicating no intermittency in the spatial dist…
Fractal structures emerge from statistical and hierarchical processes in urban development or network evolution. In a class of efficient and robust geographical networks, we derive the size distribution of layered areas, and estimate the fractal dimension by using the distribution without huge computations. This method can be applied to self-similar tilings based on a stochastic process.
Morphologies of urban patterns display multifractal scaling. However, what data should be used to represent an urban pattern and its scaling? Here, we calculated Renyi's generalized dimensions (RGD) spectra using data corresponding to different urban modalities including urban land cover, urban impervious surface, population density, and street intersection points. All data are circa 2010 and we calculated their RGD spectra in six urbanized areas located across the United States. We calculated the RGD spectra by using Hill's numbers rather than statistical moments which leads to a clear interp…
The relationships between urban area and population size have been empirically demonstrated to follow the scaling law of allometric growth. This allometric scaling is based on exponential growth of city size and can be termed "exponential allometry", which is associated with the concepts of fractals. However, both city population and urban area comply with the course of logistic growth rather than exponential growth. In this paper, I will present a new allometric scaling based on logistic growth to solve the abovementioned problem. The logistic growth is a process of replacement dynamics. Defi…
Zipf's law is one the most conspicuous empirical facts for cities, however, there is no convincing explanation for the scaling relation between rank and size and its scaling exponent. Based on the idea from general fractals and scaling, this paper proposes a dual competition hypothesis of city develop to explain the value intervals and the special value, 1, of the power exponent. Zipf's law and Pareto's law can be mathematically transformed into one another. Based on the Pareto distribution, a frequency correlation function can be constructed. By scaling analysis and multifractals spectrum, th…
A new angle of view is proposed to find the simple rules dominating complex systems and regular patterns behind random phenomena such as cities. Hierarchy of cities reflects the ubiquitous structure frequently observed in the natural world and social institutions. Where there is a hierarchy with cascade structure, there is a rank-size distribution following Zipf's law, and vice versa. The hierarchical structure can be described with a set of exponential functions that are identical in form to Horton-Strahler's laws on rivers and Gutenberg-Richter's laws on earthquake energy. From the exponenti…
The recent quantitative approaches for studing several aspects of urban life and infrastructure have shown that scale properties allow to understand many features of urban infrastructure and of human activity in cities. In this work, an analysis based on the complexity of cities network is performed for data on Brazilian cities firearms homicides. Due to the diversity in Brazilian population and cities, this is an interesting test for theories recently proposed. The superlinear power-law behavior for number of homicides as a function of city population, with exponent $β=1.15$ is obtained. The …
Understanding the morphology of an urban system is an important step toward unveiling the dynamical processes of its growth and development. At the foundation of every urban system, transportation system is undeniably a crucial component in powering the life of the entire urban system. In this work, we study the spatial pattern of $73$ cities across the globe by analysing the distribution of public transport points within the cities. The analysis reveals that different spatial distributions of points could be classified into four groups with distinct features, indicating whether the points are…
The urban road networks of the 20 largest German cities have been analysed, based on a detailed database providing the geographical positions as well as the travel-times for network sizes up to 37,000 nodes and 87,000 links. As the human driver recognises travel-times rather than distances, faster roads appear to be 'shorter' than slower ones. The resulting metric space has an effective dimension d>2, which is a significant measure of the heterogeneity of road speeds. We found that traffic strongly concentrates on only a small fraction of the roads. The distribution of vehicular flows over the…
Cities can be characterised and modelled through different urban measures. Consistency within these observables is crucial in order to advance towards a science of cities. Bettencourt et al have proposed that many of these urban measures can be predicted through universal scaling laws. We develop a framework to consistently define cities, using commuting to work and population density thresholds, and construct thousands of realisations of systems of cities with different boundaries for England and Wales. These serve as a laboratory for the scaling analysis of a large set of urban indicators. T…
Human flow in cities indicates social activity and can reveal urban spatial structures based on human behaviours for relevant applications. Scalar potential is a mathematical concept, and if successfully introduced, it can provide an intuitive perspective of human flow. However, the definition of such a potential to the origin-destination flow matrix and determination of its plausibility remain unsolved. Here, we apply Hodge-Kodaira decomposition, in which a matrix is uniquely decomposed into a potential-driven (gradient) flow and a curl flow. We depict the potential landscapes in cities due t…
The scaling exponent of a hierarchy of cities used to be regarded as a fractal parameter. The Pareto exponent was treated as the fractal dimension of size distribution of cities, while the Zipf exponent was treated as the reciprocal of the fractal dimension. However, this viewpoint is not exact. In this paper, I will present a new interpretation of the scaling exponent of rank-size distributions. The ideas from fractal measure relation and the principle of dimension consistency are employed to explore the essence of Pareto's and Zipf's scaling exponents. The Pareto exponent proved to be a rati…
In recent times, the domain of network science has become extremely useful in understanding the underlying structure of various real-world networks and to answer non-trivial questions regarding them. In this study, we rigourously analyze the statistical properties of the bus networks of six major Indian cities as graphs in L- and P-space, using tools from network science. Although public transport networks, such as airline and railway networks have been extensively studied, a comprehensive study on the structure and growth of bus networks is lacking. In India, where bus networks play an import…
Beijing Jiaotong University（北京交通大学）；Leibniz Institute of Ecological Urban and Regional Development (IOER)（莱布尼茨生态城市与区域发展研究所）；Federal University of Lavras (UFLA)（拉夫拉斯联邦大学）；University College London（伦敦大学学院）
The results of transportation infrastructure network analyses have been used to analyze complex networks in a topological context. However, most modeling approaches, including those based on complex network theory, do not fully account for real-life traffic patterns and may provide an incomplete view of network functions. This study utilizes trip data obtained from the Beijing Subway System to characterize individual passenger movement patterns. A directed weighted passenger flow network was constructed from the subway infrastructure network topology by incorporating trip data. The passenger f…
