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The conventional concept of geographical space is mainly referred to actual space based on landscape, maps, and remote sensing images. However, this notion of space is not enough to interpret different types of fractal dimension of cities. The fractal dimensions derived from Zipf's law and time series analysis do not belong to the traditional geographical space. Based on the nature of the datasets, the urban allometry can be divided into three types: longitudinal allometry indicating time, transversal allometry indicating hierarchy, and isoline allometry indicating space. According to the principle of dimension consistency, an allometric scaling exponent must be a ratio of one fractal dimension to another. From abovementioned three allometric models, we can derive three sets of fractal dimension. In light of the three sets of fractal dimension and the principle of dimension uniqueness, urban geographical space falls into three categories, including the real space based on isoline allometry and spatial distribution, the phase space based on longitudinal allometry and time series, and order space based on transversal allometry and rank-size distribution. The generalized space not only helps to explain various fractal dimensions of cities, but also can be used to develop new theory and methods of geospatial analysis., Comment: 16 pages, 4 figures, 4 tables

City is proved to be a scale-free phenomenon, and spatial autocorrelation is often employed to analyze spatial redundancy of cities. Unfortunately, spatial analysis results deviated practical requirement in many cases due to fractal nature of cities. This paper is devoted to revealing the internal relationship between the scale dependence of Moran's I and fractal scaling. Mathematical reasoning and empirical analysis are employed to derive and test the model on the scale dependence of spatial autocorrelation. The data extraction way for fractal dimension estimation is box-counting method, and parameter estimation relies on the least squares regression. In light of the locality postulate of spatial correlation and the idea of multifractals, a power law model on Moran's I changing with measurement scale is derived from the principle of recursive subdivision of space. The power exponent is proved to be a function of fractal dimension. This suggests that the numerical relationship between Moran's I and fractal dimension can be established through the scaling process of granularity. An empirical analysis is made to testify the theoretical model. It can be concluded that spatial autocorrelation of urban built-up area has no characteristic scale in many cases, and urban spatial analysis need new thinking., Comment: 25 pages, 6 figures, 5 tables

A time series of fractal dimension values of urban form can form a fractal dimension curve and reflects urban growth. In many cases, the fractal dimension curves of cities can be modeled with logistic function, which in turn can be used to make prediction analysis and stage division studies of urban evolution. Although there is more than one method available, it is difficult for many scholars to estimate the capacity parameter value in a logistic model. This paper shows a nonlinear autoregressive approach to estimating parameter values of logistic growth model of fractal dimension curves. The process is as follows. First, differentiating logistic function in theory with respect to time yields a growth rate equation of fractal dimension. Second, discretizing the growth rate equation yields a nonlinear autoregressive equation of fractal dimension. Third, applying the least square calculation to the nonlinear autoregressive equation yields partial parameter values of the logistic model. Fourth, substituting the preliminarily estimated results into the logistic models and changing it into a linear form, we can estimate the other parameter values by linear regression analysis. Finally, a practical logistic model of fractal dimension curves is obtained. The approach is applied the Baltimore's and Shenzhen's fractal dimension curves to demonstrate how to make use of it. This study provides a simple and effective method for estimating logistic model parameters, and it can be extended to the logistic models in other fields., Comment: 22 pages, 3 figures, 9 tables

Fractal dimension is an effective scaling exponent of characterizing scale-free phenomena such as cities. Urban growth can be described with time series of fractal dimension of urban form. However, how to explain the factors behind fractal dimension sequences that affect fractal urban growth remains a problem. This paper is devoted to developing a method of logistic regression modeling, which can be employed to find the influencing factors of urban growth and rank them in terms of importance. The logistic regression model comprises three components. The first is a linear function indicating the relationship between time dummy and influencing variables. The second is a logistic function linking fractal dimension and time dummy. The third is a ratio function representing normalized fractal dimension. The core composition is the logistic function that implies the dynamics of spatial replacement. The logistic regression modeling can be extended to other spatial replacement phenomena such as urbanization, traffic network development, and technology innovation diffusion. This study contributes to the development of quantitative analysis tools based on the combination of fractal geometry and conventional mathematical methods., Comment: 15 pages, 2 figures, 1 table

We propose a new method to accelerate online Mixed Integer Optimization with Pre-trained machine learning models (PreMIO). The key component of PreMIO is a multi-variable cardinality branching procedure that splits the feasible region with data-driven hyperplanes, which can be easily integrated into any MIP solver with two lines of code. Moreover, we incorporate learning theory and concentration inequalities to develop a straightforward and interpretable hyper-parameter selection strategy for our method. We test the performance of PreMIO by applying it to state-of-the-art MIP solvers and running numerical experiments on both classical OR benchmark datasets and real-life instances. The results validate the effectiveness of our proposed method.

Fractal geometry proved to be an effective mathematical tool for exploring real geographical space based on digital maps and remote sensing images. Whether the fractal theory tool can be applied to abstract geographical space has not been reported. An abstract space can be defined by multivariable distance metrics, which is frequently met in scientific research. Based on the ideas from fractals, this paper is devoted to developing correlation dimension analysis method for generalized geographical space by means of mathematical derivation and empirical analysis. Defining a mathematical distance or statistical distance, we can construct a generalized correlation function. If the relationship between correlation function and correlation lengths follows a power law, the power exponent can be demonstrated to associate with correlation dimension. Thus fractal dimension can be employed to analyze the structure and nature of generalized geographical space. This suggests that fractal geometry can be generalized to explore scale-free abstract geographical space. The theoretical model was proved mathematically, and the analytical method was illustrated by using observational data. This research is helpful to expand the application of fractal theory in geographical analysis, and the results and conclusions can be extended to other scientific fields., Comment: 26 pages, 8 figures, 5 tables

Moran's index is a basic measure of spatial autocorrelation, which has been applied to varied fields of both natural and social sciences. A good measure should have clear boundary values or critical value. However, for Moran's index, both boundary values and critical value are controversial. In this paper, a novel method is proposed to derive the boundary values of Moran's index. The key lies in finding conditional extremum based on quadratic form of defining Moran's index. As a result, two sets of boundary values are derived naturally for Moran's index. One is determined by the eigenvalues of spatial weight matrix, and the other is determined by the quadratic form of spatial autocorrelation coefficient (-1

Based on standardized vector and globally normalized weight matrix, Moran's index of spatial autocorrelation analysis has been expressed as a formula of quadratic form. Further, based on this formula, an inner product equation and outer product equation of the standardized vector can be constructed for Moran's index. However, the theoretical foundations and application direction of these equations are not yet clear. This paper is devoted to exploring the inner and outer product equations of Moran's index. The methods include mathematical derivation and empirical analysis. The results are as follows. First, based on the inner product equation, two spatial autocorrelation models can be constructed. One bears constant terms, and the other bear no constant term. The spatial autocorrelation models can be employed to calculate Moran's index by regression analysis. Second, the inner and outer product equations can be used to improve Moran's scatterplot. The normalized Moran's scatterplot can show more geospatial information than the conventional Moran's scatterplot. A conclusion can be reached that the spatial autocorrelation models are useful spatial analysis tools, complementing the uses of spatial autocorrelation coefficient and spatial autoregressive models. These models are helpful for understanding the boundary values of Moran's index and spatial autoregressive modeling process., Comment: 23 pages, 2 figures, 4 tables. arXiv admin note: text overlap with arXiv:2203.13188

Spatial autocorrelation measures such as Moran's index can be expressed as a pair of equations based on a standardized size variable and a globally normalized weight matrix. One is based on inner product, and the other is based on outer product of the size variable. The inner product equation is actually a spatial autocorrelation model. However, the theoretical basis of the inner product equation for Moran's index is not clear. This paper is devoted to revealing the antecedents and consequences of the inner product equation of Moran's index. The method is mathematical derivation and empirical analysis. The main results are as follows. First, the inner product equation is derived from a simple spatial autoregressive model, and thus the relation between Moran's index and spatial autoregressive coefficient is clarified. Second, the least squares regression is proved to be one of effective approaches for estimating spatial autoregressive coefficient. Third, the value ranges of the spatial autoregressive coefficient can be identified from three angles of view. A conclusion can be drawn that a spatial autocorrelation model is actually an inverse spatial autoregressive model, and Moran's index and spatial autoregressive models can be integrated into the same framework through inner product and outer product equations. This work may be helpful for understanding the connections and differences between spatial autocorrelation measurements and spatial autoregressive modeling., Comment: 25 pages, 2 figures, 2 tables

The local indicators of spatial association (LISA) are significant measures for spatial autocorrelation analysis. However, there is an inadvertent fault in Anselin's mathematical processes so that the local Moran and Geary indicators do not satisfy his second basic requirement, i.e., the sum of the local indicators is proportional to a global indicator. Based on Anselin's original intention, this paper is devoted to reconstructing the calculation formulae of the local Moran indexes and Geary coefficients through mathematical derivation and empirical evidence. Two sets of LISAs were clarified by mathematical reasoning. One set of LISAs is based on no normalized weights and centralized variable (MI1 and GC1), and the other set is but the second the set cannot. Then, the third set of LISA was proposed, treated as canonical forms (MI3 and GC3). The local Moran indexes are based on global normalized weights and standardized variable based on population standard deviation, while the local Geary coefficients are based on global normalized weights and standardized variable based on sample standard deviation. This set of LISAs satisfies the second requirement of based on row normalized weights and standardized variable (MI2 and GC2). The results show that the first set of LISAs satisfy Anselin's second requirement,Anselin's. The observational data of city population and traffic mileage in Beijing-Tianjin-Hebei region of China were employed to verify the theoretical results. This study helps to clarify the misunderstandings about LISAs in the field of geospatial analysis., Comment: 32 pages, 2 figures, 9 tables

Spatial autocorrelation analysis is the basis for spatial autoregressive modeling. However, the relationships between spatial correlation coefficients and spatial regression models are not yet well clarified. The paper is devoted to explore the deep structure of spatial regression coefficients. By means of mathematical reasoning, a pair of formulae of canonical spatial regression coefficients are derived from a general spatial regression model based on standardized variables. The spatial auto- and lag-regression coefficients are reduced to a series of statistic parameters and measurements, including conventional regressive coefficient, Pearson correlation coefficient, Moran's indexes, spatial cross-correlation coefficients, and the variance of prediction residuals. The formulae show determinate inherent relationships between spatial correlation coefficients and spatial regression coefficients. New finding is as below: the spatial autoregressive coefficient mainly depends on the Moran's index of the independent variable, while the spatial lag-regressive coefficient chiefly depends on the cross-correlation coefficient of independent variable and dependent variable. The observational data of an urban system in Beijing, Tianjin, and Hebei region of China were employed to verify the newly derived formulae, and the results are satisfying. The new formulae and their variates are helpful for understand spatial regression models from the perspective of spatial correlation and can be used to assist spatial regression modeling., Comment: 24 pages, 2 figures, 7 tables

Passenger flows in a traffic network reflect spatial interaction patterns in an urban systems. Gravity models can be employed to quantitatively describe and predict spatial flows. However, how to model passenger flows and reveal the deep structure of urban and traffic networks in the case of missing partial data is still a problem to be solved. This paper is devoted to characterizing the interurban passenger flows in the Beijing-Tianjin-Hebei region of China by means of dual gravity models and Tencent location big data. The method of parameter estimation is the least squares regression. The main results are as follows. First, both railway and highway passenger flows can be effectively described by the dual gravity model. A small part of missing spatial data can be made up by the predicted values. Second, the fractal properties of traffic flows can be revealed. The railway passenger flows more follow gravity scaling law than the highway passenger flows. Third, the prediction residuals indicate the changing trend of interurban connections in the study area in recent years. The center of gravity of spatial dynamics seems to shift from the Beijing-Tianjin-Tangshan triangle to the Beijing-Baoding-Shijiazhuang axis. A conclusion can be reached that the dual gravity models is an effective tools of for analyzing spatial structures and dynamics of traffic networks and flows. Moreover, the models provide a new approach to estimate fractal dimension of traffic network and spatial flow patterns., Comment: 29 pages, 7 figures, 4 tables

The \b{eta} index is one of important measurements reflecting development level of traffic networks. However, how to explain and predict the \b{eta} index growth for a geographical region is a pending problem. With the help of mathematical reasoning and empirical analysis, this paper is devoted to modeling the growing curve of \b{eta} index. A new measurement termed {\delta} index is introduced and a set of new models are constructed. Suppose there is a nonlinear relation between human settlements and roads. A pair differential equations are built for describing the nonlinear dynamics of traffic networks. A logistic function of \b{eta} index growth is derived from the two spatial dynamic equations. On the other hand, based on verified empirical models about urbanization level, economic development level, and \b{eta} index of traffic network, a Boltzmann equation of \b{eta} index growth can be derived. Normalizing the \b{eta} index, Boltzmann equation will become logistic function. This lends an indirect support to the theoretical model of \b{eta} index from the prospective of positive studies. The models proposed in this work provided new approaches for explanation and prediction of spatio-temporal evolution of traffic networks., Comment: 21 pages, 6 figures, 2 tables

A pair of fractal gravity models can be derived from the spatial interaction models based on entropy maximizing principle and allometric scaling law. The models can be expressed as dual form in mathematics and are important for analyzing and predicting spatial flows in network of cities. However, the dual relationship of urban gravity parameters has been an empirical relationship for a long time and lacks theoretical proof. This paper is devoted to proving the duality of fractal gravity models by means of ideas from scaling invariance. The results show that a fractal gravity model can be derived from its dual form. The observational data of interurban spatial flows of Beijing-Tianjin-Hebei region of China are employed to make a case study, lending a further support to the theoretical derivation. A conclusion can be reached that the duality of gravity models rests with scaling symmetry of fractal structure. This work may be helpful for understanding the theoretical essence and application direction of spatial interaction modeling., Comment: 19 pages, 1 figure, 6 tables

The curves of scaling behavior is a significant concept in fractal dimension analysis of complex systems. However, the underlying rationale of this kind of curves for fractal cities is not yet clear. This paper is devoted to researching a set of basic problems of the scaling behavior curves in urban studies by using mathematical reasoning and empirical analysis. The main findings are as follows. First, the formula of scaling behavior curves is derived from a fractal model based on hierarchical structure of urban systems. Second, the relationships between the formula of scaling behavior curves and similarity dimension and scale-dependent fractal dimension are revealed. Third, according to the fractal dimension measurement methods, the scaling behavior curves are divided into two different types. Fourth, empirically, 1-dimensional spatial autocorrelation function of scaling behavior curves can be employed to reveal the basic property of scaling behavior curves. In comparison, scaling behavior curves are more sensitive to power law relations than the fractal distribution functions. The scaling curves can be utilized to evaluate fractal development extent of urban systems and identify scaling ranges of fractal cities. In positive studies, the curves can be used to help distinguish the boundaries between urban areas and rural areas and self-affine fractal structure behind the dynamical process of spatial correlation., Comment: 25 pages, 10 figures, 2 tables

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, and terminal slow growth. The three dividing points are 0.2113Dmax, 0.5Dmax, and 0.7887Dmax, where Dmax is the capacity of fractal dimension. When the fractal dimension reaches half of its capacity value, 0.5Dmax, the urban growth rate reaches its peak. (2) Based on the curve of fractal dimension odds, urban growth can also be divided into four stages: initial slow filling, accelerated fast filling, decelerated fast filling, terminal slow filling. The three dividing points are 0.2113Zmax, 0.5Zmax, and 0.7887Zmax, where Zmax=/(2-Dmax) denotes the capacity of fractal dimension odds (Dmax<2). Empirical analyses show that the first scheme based fractal dimension is suitable for the young cities and the second scheme based on fractal dimension odds can be applied to mature cities. A conclusion can be reached that logistic function is one of significant model for the stage division of urban growth based on fractal parameters of cities. The results of this study provide a new way of understanding the features and mechanism of urban phase transition., Comment: 33 pages, 11 figures, 6 tables

The spatial diffusion of epidemic disease follows distance decay law in geography, but different diffusion processes may be modeled by different mathematical functions under different spatio-temporal conditions. This paper is devoted to modeling spatial diffusion patterns of COVID-19 stemming from Wuhan city to Hubei province. The methods include gravity and spatial auto-regression analyses. The local gravity model is derived from allometric scaling and global gravity model, and then the parameters of the local gravity model are estimated by observational data and linear regression. The main results are as below. The local gravity model based on power law decay can effectively describe the diffusion patterns and process of COVID-19 in Hubei Province, and the goodness of fit of the gravity model based on negative exponential decay to the observation data is not satisfactory. Further, the goodness of fit of the model to data entirely became better and better over time, the size elasticity coefficient increases first and then decreases, and the distance attenuation exponent decreases first and then increases. Moreover, the significance of spatial autoregressive coefficient in the model is low, and the confidence level is less than 80%. The conclusions can be reached as follows. (1) The spatial diffusion of COVID-19 of Hubei bears long range effect, and the size of a city and the distance of the city to Wuhan affect the total number of confirmed cases. (2) Wuhan direct transmission is the main process in the spatial diffusion of COVID-19 in Hubei at the early stage, and the horizontal transmission between regions is not significant. (3) The effect of spatial isolation measures taken by Chinese government against the transmission of COVID-19 is obvious. This study suggests that the role of gravity should be taken into account to prevent and control epidemic disease., Comment: 24 pages, 3 figures, 5 tables

Urban population density provides a good perspective for understanding urban growth and socio-spatial dynamics. Based on sub-district data of the five times of national population censuses in 1964, 1982, 1990, 2000, and 2010, this paper is devoted to making analyses of urban growth and the spatial restructuring of population in the city of Hangzhou, China. Research methods are based on mathematical modeling and field investigation. The modeling result shows that the negative exponential function and the power-exponential function can be well fitted to Hangzhou's observational data of urban density. The negative exponential model reflect the expected state, while the power-exponential model reflects the real state of urban density distribution. The parameters of these models are linearly correlated to the spatial information entropy of population distribution. The density gradient in the negative exponential function flattened in the 1990s and 2000s is closely related to the development of suburbanization. In terms of investigation materials and the changing trend of model parameters, we can reveal the spatio-temporal features of Hangzhou's urban growth. The main conclusions can be reached as follows. The policy of reformation and opening-up and the establishment of a market economy improved the development mode of Hangzhou. As long as a city has a good social and economic environment, it will automatically tend to the optimal state through self-organization., Comment: 29 pages,4 figures, 6 tables

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 networks of 10 Chinese cities are taken as cases for positive studies. Fast Fourier transform and least squares regression analysis are employed to calculate spectral exponents. The results show that the wave-spectral density distribution of all these urban traffic networks follows scaling law, and the spectral scaling exponents can be converted to fractal dimension values. Using the fractal parameters, we can make spatial analyses for the geographical signals. The analytical process can be generalized to temporal signal analyses. The wave-spectrum scaling methods can be applied to both self-similar fractal signals and self-affine fractal signals in the geographical world., Comment: 22 pages, 7 figures, 4 tables

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 southern China. The urban region is divided into four subareas, ArcGIS technology, box-counting method is adopted to extract spatial datasets, and the least squares regression is employed to estimate fractal parameters. The results show that: (1) The urban form of Shenzhen city has clear fractal structure, but fractal dimension values of different subareas are different; (2) The fractal dimension growth curves of all the four study areas can only be modeled by the common logistic function, and the goodness of fit increases over time; (3) The peak of urban growth in Shenzhen had passed before 1986, the fractal dimension growth is approaching its maximum capacity. Conclusions can be reached that the urban form of Shenzhen bears characteristics of multifractals, the fractal structure has been becoming better gradually through self-organization, but its land resources are reaching the limits of growth. The fractal dimension curves of Shenzhen's urban growth are similar to those of European and American cities, but differ from those of the cities in northern China. This suggests that there is subtle different dynamic mechanisms of city development between northern and southern China., Comment: 24 pages,7 figures, 3 tables

Revealing spatiotemporal evolution regularity in the spatial diffusion of epidemics is helpful for preventing and controlling the spread of epidemics. Based on the real-time COVID-19 datasets by prefecture-level cities, this paper is devoted to exploring the multifractal scaling in spatial diffusion pattern of COVID-19 pandemic and its evolution characteristics in Chinese mainland. The ArcGIS technology and box-counting method are employed to extract spatial data and the least square regression based on rescaling probability (miu-weight method) is used to calculate fractal parameters. The results show multifractal distribution of COVID-19 pandemic in China. The generalized correlation dimension spectrums are inverse S-shaped curves, but the fractal dimension values significantly exceed the Euclidean dimension of embedding space when moment order q<<0. The local singularity spectrums are asymmetric unimodal curves, which slant to right. The fractal dimension growth curves are shown as quasi S-shaped curves. From these spectrums and growth curves, the main conclusions can be drawn as follows: First, self-similar patterns developed in the process of COVID-19 pandemic, which seem be dominated by multifractal scaling law. Second, the spatial pattern of COVID-19 across China can be characterized by global clustering with local disordered diffusion. Third, the spatial diffusion process of COVID-19 in China experienced four stages, i.e., initial stage, the rapid diffusion stage, the hierarchical diffusion stage, and finally the contraction stage. This study suggests that multifractal theory can be utilized to characterize spatio-temporal diffusion of COVID-19 pandemic, and the case analyses may be instructive for further exploring natural laws of spatial diffusion., Comment: 28 pages,6 figures, 5 tables

The rank-size distribution of cities follows Zipf's law, and the Zipf scaling exponent often tends to a constant 1. This seems to be a general rule. However, a recent numerical experiment shows that there exists a contradiction between the Zipf exponent 1 and high urbanization level in a large population country. In this paper, mathematical modeling, computational analysis, and the method of proof by contradiction are employed to reveal the numerical relationships between urbanization level and Zipf scaling exponent. The main findings are as follows. (1) If Zipf scaling exponent equals 1, the urbanization rate of a large populous country can hardly exceed 50%. (2) If Zipf scaling exponent is less than 1, the urbanization level of large populous countries can exceeds 80%. A conclusion can be drawn that the Zipf exponent is the control parameter for the urbanization dynamics. In order to improve the urbanization level of large population countries, it is necessary to reduce the Zipf scaling exponent. Allometric growth law is employed to interpret the change of Zipf exponent, and scaling transform is employed to prove that different definitions of cities do no influence the above analytical conclusion essentially. This study provides a new way of looking at Zipf's law of city-size distribution and urbanization dynamics., Comment: 29 pages, 2 figures, 8 tables

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 analogy with fractal dimension growth models of cities, a pair of entropy increase models can be derived and a set of entropy-based measurements can be constructed to describe urban growing process and patterns. First, logistic function and Boltzmann equation are utilized to model the entropy increase curves of urban growth. Second, a series of indexes based on spatial entropy are used to characterize urban form. Further, multifractal dimension spectrums are generalized to spatial entropy spectrums. Conclusions are drawn as follows. Entropy and fractal dimension have both intersection and different spheres of application to urban research. Thus, for a given spatial measurement scale, fractal dimension can often be replaced by spatial entropy for simplicity. The models and measurements presented in this work are significant for integrating entropy and fractal dimension into the same framework of urban spatial analysis and understanding spatial complexity of cities., Comment: 29 pages, 4 figure, 6 tables

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 multifractal parameters. The results showed multifractal structure of urban street networks. The global multifractal dimension spectrums are inverse S-shaped curves, while the local singularity spectrums are asymmetric unimodal curves. If the moment order q approaches negative infinity, the generalized correlation dimension will seriously exceed the embedding space dimension 2, and the local fractal dimension curve displays an abnormal decrease for most cities. The scaling relation of local fractal dimension gradually breaks if the q value is too high, but the different levels of the network always keep the scaling reflecting singularity exponent. The main conclusions are as follows. First, urban street networks follow multifractal scaling law, and scaling precedes local fractal structure. Second, the patterns of traffic networks take on characteristics of spatial concentration, but they also show the implied trend of spatial deconcentration. Third, the development space of central area and network intensive areas is limited, while the fringe zone and network sparse areas show the phenomenon of disordered evolution. This work may be revealing for understanding and further research on complex spatial networks by using multifractal theory., Comment: 32 pages, 9 figures, 5 tables

The law of allometric growth is one of basic rules for understanding urban evolution. The general form of this law is allometric scaling law. However, the deep meaning and underlying rationale of the scaling exponents remain to be brought to light. In this paper, the theories of linear algebra and regression analysis are employed to reveal the mathematical and statistic essence of allometric scaling exponents. Suppose that the geometric measure relations between a set of elements in an urban system follow the allometric growth law. An allometric scaling exponent is proved to equal in theory to the ratio of the standard deviation of one logarithmic measure to the standard deviation of another logarithmic measure. In empirical analyses based on observational data, the scaling exponent is equal to the product between the standard deviation ratio and the corresponding Pearson correlation coefficient. The mathematical derivation results can be verified by empirical analysis: the scaling exponent values based on the standard deviation ratios are completely identical to those based on the conventional method. This finding can be generalized to city fractals and city size distribution to explain fractal dimensions of urban space and Zipf scaling exponent of urban hierarchy. A conclusion can be reached that scaling exponents reflect the ratios of characteristic lengths. This study may be helpful for comprehending scaling from a new perspective and the connections and distinctions between scaling and characteristic scales., Comment: 23 pages, 3 figures, 5 tables

The traditional concept of space in geography is based on the notion of distance. Where there is a spatial analysis, there is a distance measurement. However, the precondition for effective distance-based space is that the geographical systems have characteristic scales. For a scale-free geographical system, the spatial structure cannot be validly described with pure distance, and thus the distance-based space is ineffective for geographical modelling. In the real geographical world, scale-free patterns and processes are everywhere. We need new notion of geographical space. Using the ideas from fractals and scaling relations, I propose a dimension-based concept of space for scale-free geographical analysis. If a geographical phenomenon bears characteristic scales, we can model it using distance measurement; if a geographical phenomenon has no characteristic scale, we will describe it using fractal dimension, which is based on the scaling relations between distance variable and the corresponding measurements. In short, geographical space fall into two types: scaleful space and scale-free space. This study shows a new way of spatial modeling and quantitative analyses for the geographical systems without characteristic scale., Comment: 22 pages, 7 tables

Spatial interaction and spatial autocorrelation are two different fields of geo-spatial analysis, revealing the internal relationship between the two fields will help to develop the theory and method of geographical analysis. This paper is devoted to deducing a system of spatial correlation analysis models from the gravity model by mathematical derivation. The main results are as follows. First, a set of potential energy measurements are derived from the gravity model. Second, a pair of correlation equations, including an inner product equation and an outer product equation, are constructed based on the quadratic form of potential energy formula. Third, a series of spatial autocorrelation statistics, including Moran's index and Getis-Ord's index are derived from the potential energy formula. Fourth, the concept of fractal dimension is introduced into spatial weight matrix. The observational data of urban systems are employed to make an empirical analysis, demonstrating the application procedure of newly derived models. A conclusion can be drawn that spatial autocorrelation is actually rooted in spatial interaction process, and an improved methodology of spatial analysis can be developed by integrating spatial autocorrelation models and gravity model into the same framework., Comment: 30 pages, 7 figure, 6 tables

Geographical research was successfully quantified through the quantitative revolution of geography. However, the succeeding theorization of geography encountered insurmountable difficulties. The largest obstacle of geography's theorization lies in scale-free distributions of geographical phenomena which exist everywhere. The first paradigm of scientific research is mathematical theory. The key of a quantitative measurement and mathematical modeling is to find a valid characteristic scale. Unfortunately, for many geographical systems, there is no characteristic scale. In this case, the method of scaling should be employed to make a spatial measurement and carry out mathematical modeling. The basic idea of scaling is to find a power exponent using the double logarithmic linear relation between a variable scale and the corresponding measurement results. The exponent is a characteristic parameter which follows a scaleful distribution and can be used to characterize the scale-free phenomena. The importance of the scaling analysis in geography is becoming more and more evident for scientists., Comment: 19 pages, 2 figures, 5 tables

Geographical phenomena fall into two categories: scaleful phenomena and scale-free phenomena. The former bears characteristic scales, and the latter has no characteristic scale. The conventional quantitative and mathematical methods can only be effectively applied to scaleful geographical phenomena rather than the scale-free geographical phenomena. In this paper, a comparison between scaleful geographical systems and scale-free geographical systems are drawn by means of simple geographical mathematical models. The main viewpoints are as below. First, the scaleful phenomena can be researched by conventional mathematical methods, while the scale-free phenomena should be studied using the theory based on scaling such as fractal geometry; Second, the scaleful phenomena belong to distance-based geo-space, while the scale-free phenomena belong to dimension-based geo-space; Third, four approaches to distinguish scale-free phenomena from scaleful phenomena are presented, including scaling transform, probability distribution, autocorrelation and partial autocorrelation functions, and ht-index. In practice, a complex geographical system usually possesses scaleful aspects and scale-free aspects. Different methodologies must be adopted for different types of geographic systems or different aspects of the same geographic system., Comment: 27 pages, 6 figures, 7 tables

A number of spatial statistic measurements such as Moran's I and Geary's C can be used for spatial autocorrelation analysis. Spatial autocorrelation modeling proceeded from the 1-dimension autocorrelation of time series analysis, with time lag replaced by spatial weights so that the autocorrelation functions degenerated to autocorrelation coefficients. This paper develops 2-dimensional spatial autocorrelation functions based on the Moran index using the relative staircase function as a weight function to yield a spatial weight matrix with a displacement parameter. The displacement bears analogy with time lag of time series analysis. Based on the spatial displacement parameter, two types of spatial autocorrelation functions are constructed for 2-dimensional spatial analysis. Then the partial spatial autocorrelation functions are derived by Yule-Walker recursive equation. The spatial autocorrelation functions are generalized to the autocorrelation functions based on Geary's coefficient and Getis' index. As an example, the new analytical framework was applied to the spatial autocorrelation modeling of Chinese cities. A conclusion can be reached that it is an effective method to build an autocorrelation function based on the relative step function. The spatial autocorrelation functions can be employed to reveal deep geographical information and perform spatial dynamic analysis, and lay the foundation for the scaling analysis of spatial correlation., Comment: 36 pages, 11 figures, 5 tables

Spatial autocorrelation coefficients such as Moran's index proved to be an eigenvalue of the spatial correlation matrixes. An eigenvalue represents a kind of characteristic length for quantitative analysis. However, if a spatial correlation is based on self-organized evolution, complex structure, and the distributions without characteristic scale, the eigenvalue will be ineffective. In this case, the single Moran index cannot lead to reliable statistic inferences. This paper is devoted to finding advisable approach to measure spatial autocorrelation for the scale-free processes of complex systems by means of mathematical reasoning and empirical analysis. Based on relative step function as spatial contiguity function, a series of ordered spatial autocorrelation coefficients are converted into the corresponding spatial autocorrelation functions. Then the mathematical relation between spatial correlation dimension and spatial autocorrelation functions is derived by decomposition of spatial autocorrelation functions. As results, a set of useful mathematical models are constructed for spatial analysis. Using these models, we can utilize spatial correlation dimension to make simple spatial autocorrelation analysis, and use spatial autocorrelation functions to make complex spatial autocorrelation analysis for geographical phenomena. This study reveals the inherent association of fractal patterns with spatial autocorrelation processes in nature and society. The work may inspire new ideas of spatial modeling and analysis for complex geographical systems., Comment: 32 pages, 8 figure, 5 table

Irregular boundary lines can be characterized by fractal dimension, which provides important information for spatial analysis of complex geographical phenomena such as cities. However, it is difficult to calculate fractal dimension of boundaries systematically when image data is limited. An approximation estimation formulae of boundary dimension based on square is widely applied in urban and ecological studies. However, the boundary dimension is sometimes overestimated. This paper is devoted to developing a series of practicable formulae for boundary dimension estimation using ideas from fractals. A number of regular figures are employed as reference shapes, from which the corresponding geometric measure relations are constructed; from these measure relations, two sets of fractal dimension estimation formulae are derived for describing fractal-like boundaries. Correspondingly, a group of shape indexes can be defined. A finding is that different formulae have different merits and spheres of application, and the second set of boundary dimensions is a function of the shape indexes. Under condition of data shortage, these formulae can be utilized to estimate boundary dimension values rapidly. Moreover, the relationships between boundary dimension and shape indexes are instructive to understand the association and differences between characteristic scales and scaling. The formulae may be useful for the pre-fractal studies in geography, geomorphology, ecology, landscape science, and especially, urban science., Comment: 28 pages, 2 figures, 9 tables

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 of fractal cities bears two typical properties. One is the scale dependence. Entropy values of urban systems always depend on the scales of spatial measurement. The other is entropy conservation. Different fractal parts bear the same entropy value. Thus entropy cannot reflect the spatial heterogeneity of fractal cities in theory. If we convert the generalized entropy into multifractal spectrums, the problems of scale dependence and entropy homogeneity can be solved to a degree for urban spatial analysis. The essence of scale dependence is the scaling in cities, and the spatial heterogeneity of cities can be characterized by multifractal scaling. This study may be helpful for the students to describe and understand spatial complexity of cities., Comment: 27 page, 9 figure, 5 tables

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 deduction and ideas from fractal theory, this paper is devoted to discussing fractals and fractal dimensions of urban landscape. The main points of this work are as follows. First, urban form can be treated as pre-fractals rather than real fractals, and fractal properties of cities are only valid within certain scaling ranges. Second, the topological dimension of city fractals based on urban area is 0, thus the minimum fractal dimension value of fractal cities is equal to or greater than 0. Third, fractal dimension of urban form is used to substitute urban area, and it is better to define city fractals in a 2-dimensional embedding space, thus the maximum fractal dimension value of urban form is 2. A conclusion can be reached that urban form can be explored as fractals within certain ranges of scales and fractal geometry can be applied to the spatial analysis of the scale-free aspects of urban morphology., Comment: 33 pages, 3 figures, 10 tables

Spatial autocorrelation and spatial interaction are two important analytical processes for geographical analyses. However, the internal relations between the two types of models have not been brought to light. This paper is devoted to integrating spatial autocorrelation analysis and spatial interaction analysis into a logic framework by means of Getis-Ord's indexes. Based on mathematical derivation and transform, the spatial autocorrelation measurements of Getis-Ord's indexes are reconstructed in a new and simple form. A finding is that the local Getis-Ord's indexes of spatial autocorrelation are equivalent to the rescaled potential energy indexes of spatial interaction theory based on power-law distance decay. The normalized scatterplot is introduced into the spatial analysis based on Getis-Ord's indexes, and the potential energy indexes are proposed as a complementary measurement. The global Getis-Ord's index proved to be the weighted sum of the potential energy indexes and the direct sum of total potential energy. The empirical analysis of the system of Chinese cities are taken as an example to illustrate the effect of the improved methods and measurements. The mathematical framework newly derived from Getis-Ord's work is helpful for further developing the methodology of geographical spatial modeling and quantitative analysis., Comment: 31 pages, 6 figures, 5 tables

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, multi-scaling fractal patterns, and self-affine fractal growth. The solution to the problem is to substitute the real fractal dimension values with comparable fractal dimensions. The main measures are as follows: First, select a proper method for a special fractal study. Second, define a proper study area for a city according to a study aim, or define comparable study areas for different cities. These suggestions may be helpful for the students who takes interest in or even have already participated in the studies of fractal cities., Comment: 27 pages, 3 figures, 8 tables

Defining an objective boundary for a city is a difficult problem, which remains to be solved by an effective method. Recent years, new methods for identifying urban boundary have been developed by means of spatial search techniques (e.g. CCA). However, the new algorithms are involved with another problem, that is, how to determine the characteristic radius of spatial search. This paper proposes new approaches to looking for the most advisable spatial searching radius for determining urban boundary. We found that the relationships between the spatial searching radius and the corresponding number of clusters take on an exponential function. In the exponential model, the scale parameter just represents the characteristic length that we can use to define the most objective urban boundary objectively. Two sets of China's cities are employed to test this method, and the results lend support to the judgment that the characteristic parameter can well serve for the spatial searching radius. The research may be revealing for making urban spatial analysis in methodology and implementing identification of urban boundaries in practice., Comment: 26 pages, 5 figures, 7 tables

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 reconstructed, the structural dimensions are derived from the textural dimensions, and the central place fractals are formulated by the k numbers and g numbers. Three structural fractal models are constructed for central place systems according to the corresponding fractal dimensions. A theoretical finding is that the classic central place models comprise Koch snowflake curve and Sierpinski space filling curve, and an inference is that the traffic principle plays a leading role in urban and rural evolution. The conclusion can be reached that the textural fractal dimensions can be converted into the structural fractal dimensions. The latter dimensions can be directly used to appraise urban and rural settlement distributions in the real world. Thus, the textural fractals can be indirectly utilized to explain the development of the systems of human settlements., Comment: 29 pages, 7 figures, 7 tables, 2 appendices

Chaos associated with bifurcation makes a new science, but the origin and essence of chaos are not yet clear. Based on the well-known logistic map, chaos used to be regarded as intrinsic randomicity of determinate dynamics systems. However, urbanization dynamics indicates new explanation about it. Using mathematical derivation, numerical computation, and empirical analysis, we can explore chaotic dynamics of urbanization. The key is the formula of urbanization level. The urbanization curve can be described with the logistic function, which can be transformed into 1-dimensional map and thus produce bifurcation and chaos. On the other hand, the logistic model of urbanization curve can be derived from the rural-urban population interaction model, and the rural-urban interaction model can be discretized to a 2-dimensional map. An interesting finding is that the 2-dimensional rural-urban coupling map can create the same bifurcation and chaos patterns as those from the 1-dimensional logistic map. This suggests that the urban bifurcation and chaos come from spatial interaction between rural and urban population rather than pure intrinsic randomicity of determinate models. This discovery provides a new way of looking at origin and essence of bifurcation and chaos. By analogy with urbanization models, the classical predator-prey interaction model can be developed to interpret the complex dynamics of the logistic map in physical and social sciences., Comment: 29 pages, 8 figures, 3 tables

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 make empirical analyses. This study may be revealing for associating 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., Comment: 30 pages, 14 figures, 4 tables

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 Euclidean and fractal geometries. The main method is mathematical derivation and empirical analysis, and the theoretical foundation is the discovery that the normalized fractal dimension is equal to the normalized entropy. Based on this finding, a set of useful spatial indexes termed dummy multifractal parameters are defined for geographical analysis. These indexes can be employed to describe both the simple distributions and complex distributions. The dummy multifractal indexes are applied to the population density distribution of Hangzhou city, China. The calculation results reveal the feature of spatio-temporal evolution of Hangzhou's urban morphology. This study indicates that fractal dimension and spatial entropy can be combined to produce a new methodology for spatial analysis of city development., Comment: 23 pages, 3 figures, 5 tables

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, and defining rural space-filling measurement by information gain, we can construct a new urban-rural interaction and coupling model. From this model, we can derive the logistic equation of fractal dimension growth strictly. This indicates that urban growth results from the unity of opposites between spatial entropy increase and information increase. In a city, an increase in spatial entropy is accompanied by a decrease in urban land availability. An inference is that urban growth struggles between the force of urban space-filling and the force of urban space-saving. This work presents a set of urban models of spatial dynamics, which help us understand urban evolution from the angle of view of entropy and fractals., Comment: 25 pages, 5 figures, 4 tables

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., Comment: 29 pages, 7 figures, 5 tables

A standard scientific study comprises two processes: one is to describe a thing, and the other is to understand how the thing works. In order to understand the principle of urban growth, a number of shape indexes are proposed to describe the size and shape of cities. However, the comparability of a shape index is often influenced by the resolution of remote sensing images or digital maps because the calculated values depend on spatial measurement scales. This paper is devoted to exploring the scaling in classical shape indexes with mathematical methods. Two typical regular fractals, Koch's island and Vicsek's figure, are employed to illustrate the property of scale dependence of many shape indexes. A set of formula are derived from the geometric measure relation to associate fractal dimension for boundary lines with shape indexes. Based on the ideas of fractals, two main problems are solved in this work. First, several different correlated shape indexes such as circularity ratios and compactness ratios are normalized and unified by substituting Feret's diameter for the major axis of urban figure. Second, three levels of scaling in shape indexes are revealed, and shape indexes are classified into three groups. The significance of this study lies in two aspects. One is helpful for realizing the deep structure of urban form, and the other is useful for clarifying the spheres of application of different types of shape indexes., Comment: 23 pages, 5 figures, 6 tables

Hierarchies can be modeled by a set of exponential functions, from which we can derive a set of power laws indicative of scaling. The solution to a scaling relation equation is always a power law. The scaling laws are followed by many natural and social phenomena such as cities, earthquakes, and rivers. This paper is devoted to revealing the power law behaviors in systems of natural cities by reconstructing the hierarchy with cascade structure. The cities of America, Britain, France, and Germany are taken as examples to make empirical analyses. The hierarchical scaling relations can be well fitted to the data points within the scaling ranges of the size and area of the natural cities. The size-number and area-number scaling exponents are close to 1, and the allometric scaling exponent is slightly less than 1. The results show that natural cities follow hierarchical scaling laws and hierarchical conservation law very well. The hierarchical scaling law proved to be derived from entropy maximization principle, and this suggests that the evolution of natural cities is dominated by entropy maximization laws. This study is helpful for scientists to understand the power law behavior in the development of cities and systems of cities., Comment: 23 pages; 6 figures; 5 tables

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 ratio of actual entropy (Mq) to the maximum entropy (Mmax) equals the ratio of actual dimension (Dq) to the maximum dimension (Dmax). The spatial entropy and fractal dimension of complex spatial systems can be converted into one another by means of functional box-counting method. The theoretical inference is verified by observational data of urban form. A conclusion is that normalized spatial entropy is equal to normalized fractal dimension. Fractal dimensions proved to be the characteristic values of entropies. In empirical studies, if the linear size of spatial measurement is small enough, a normalized entropy value is infinitely approximate to the corresponding normalized fractal dimension value. Based on the theoretical result, new spatial measurements of urban space filling can be defined, and multifractal parameters can be generalized to describe both simple systems and complex systems., Comment: 28 pages; 3 figures; 9 tables

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 that the quadratic Boltzmann equation and quadratic logistic function can be used to characterize how the fractal dimension of the urban land-use pattern of Beijing increases in the course of time. The models are also suitable for many cities in the north of China. In order to convert the empirical models into theoretical models, we attempt to construct a model of spatial replacement dynamics of urban evolution, from which the logistic model of urban fractal dimension growth can be derived. The models can be utilized to predict the rate and upper limitation of Chinese urban growth. In particular, the models can be employed to reveal the similarities and differences between the fractal growth of Chinese cities and that of the cities in western countries., Comment: 37 pages, 5 figures, 6 tables

The law of allometric scaling based on Zipf distributions can be employed to research hierarchies of cities in a geographical region. However, the allometric patterns are easily influenced by random disturbance from the noises in observational data. In theory, both the allometric growth law and Zipf's law are related to the hierarchical scaling laws associated with fractal structure. In this paper, the scaling laws of hierarchies with cascade structure are used to study Chinese cities, and the method of R/S analysis is applied to analyzing the change trend of the allometric scaling exponents. The results show that the hierarchical scaling relations of Chinese cities became clearer and clearer from 1991 to 2014 year; the global allometric scaling exponent values fluctuated around 0.85, and the local scaling exponent approached to 0.85. The Hurst exponent of the allometric parameter change is greater than 0.5, indicating persistence and a long-term memory of urban evolution. The main conclusions can be reached as follows: the allometric scaling law of cities represents an evolutionary order rather than an invariable rule, which emerges from self-organized process of urbanization, and the ideas from allometry and fractals can be combined to optimize spatial and hierarchical structure of urban systems in future city planning., Comment: 28 pages, 10 figures, 5 tables

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 representative fractal dimensions in the multifractal dimension set are utilized to make empirical analyses of urban form of two cities. The results show that the entropy values are not determinate, but the fractal dimension value is certain; if the linear size of boxes is small enough (e.g., <1/25), the linear correlation between entropy and fractal dimension is clear. Further empirical analysis indicates that fractal dimension is close to the characteristic values of spatial entropy. This suggests that the physical meaning of fractal dimension can be interpreted by the ideas from entropy and scales and the conclusion is revealing for future spatial analysis of cities. Key words: fractal dimension; entropy; mutlifractals; scaling; urban form; Chinese cities, Comment: 26 pages, 7 figures, 8 tables

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 Clark's model indicating the characteristic radius of cities is not a real constant, but depends on the urban field we defined. So the exponential model suggests local fractal structure with two kinds of fractal parameters. The parameters can be used to characterize urban space filling, spatial correlation, self-affine properties, and self-organized evolution. The case study of the city of Hangzhou, China, is employed to verify the theoretical inference. Based on the empirical analysis, a three-ring model of cities is presented and a city is conceptually divided into three layers from core to periphery. The scaling region and non-scaling region appear alternately in the city. This model may be helpful for future urban studies and city planning.