An allometric scaling relation based on logistic growth of cities

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 above mentioned problem. The logistic growth is a process of replacement dynamics. Defining a pair of replacement quotients as new measurements, which are functions of urban area and population, we can derive an allometric scaling relation from the logistic processes of urban growth, which can be termed “logistic allometry”. The exponential allometric relation between urban area and population is the approximate expression of the logistic allometric equation when the city size is not large enough. The proper range of the allometric scaling exponent value is reconsidered through the logistic process. Then, a medium-sized city of Henan Province, China, is employed as an example to validate the new allometric relation. The logistic allometry is helpful for further understanding the fractal property and self-organized process of urban evolution in the right perspective.     

广州市人口密度分布及演化模型研究

根据广州市第三、四、五次人口普查资料,运用GIS技术对空间数据进行提取,然后对广州城市人口密度分布进行数学模拟和比较。结果表明,1982年以来广州市人口密度呈现典型的Clark分布,而且随着时间的推进,其拟合程度越来越高,今后将逐步向Newling模型演变。     

The distance-decay function of geographical gravity model: Power law or exponential law?

The distance-decay function of the geographical gravity model is originally an inverse power law, which suggests a scaling process in spatial interaction. However, the distance exponent of the model cannot be reasonably explained with the ideas from Euclidean geometry. This results in a dimension dilemma in geographical analysis. Consequently, a negative exponential function was used to replace the inverse power function to serve for a distance-decay function. But a new puzzle arose that the exponential-based gravity model goes against the first law of geography. This paper is devoted for solving these kinds of problems by mathematical reasoning and empirical analysis. New findings are as follows. First, the distance exponent of the gravity model is demonstrated to be a fractal dimension using the geometric measure relation. Second, the similarities and differences between the gravity models and spatial interaction models are revealed using allometric relations. Third, a four-parameter gravity model possesses a symmetrical expression, and we need dual gravity models to describe spatial flows. The observational data of China's cities and regions (29 elements indicative of 841 data points) in 2010 are employed to verify the theoretical inferences. A conclusion can be reached that the geographical gravity model based on power-law decay is more suitable for analyzing large, complex, and scale-free regional and urban systems. This study lends further support to the suggestion that the underlying rationale of fractal structure is entropy maximization. Moreover, it suggests that many dimensional dilemmas of spatial modeling can be solved using the concepts from fractal geometry.     

长吉图地区城镇体系的分形研究

长吉图开发开放先导区是吉林省发展的又一契机,是国家批准实验的新区,经济区和实验区当中唯一给予我国延边开发开放的重要区域地位,并赋予了先行先试权.运用分形理论对长吉图地区15个城镇自2002年-2010年九年间的城镇规模体系进行了系统定量的分形研究.作出了长吉图地区首位度和序位—规模双对数坐标图,用首位度,4城市指数,11城市指数,q值,D值进行分析研究,得出长吉图开发开放先导区城镇规模分形特征:1)长吉图地区城镇规模属于分形结构;2)城镇规模分布比较集中,中小城市不够发育,严重缺少过渡城市;3)首位分布有明显趋势,首位城市发展速度快于中小城市,城市规模出现发展不均衡.明确长吉图城镇体系在发展中的问题,据此提出城镇规模优化的合理性建议.     

A set of formulae on fractal dimension relations and its application to urban form

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 dimension and vice versa. Second, I derive the proper scales of fractal dimension: the form dimension comes between 1.5 and 2, and the boundary dimension comes between 1 and 1.5. Third, I derive three form dimension values with special geometric meanings. The first is 4/3, the second is 3/2, and the third is 1 + 2^1/2/2 ≈ 1.7071. The fractal dimension relation formulae are applied to China’s cities and the cities of the United Kingdom, and the computations are consistent with the theoretical expectation. The formulae are useful in the fractal dimension estimation of urban form, and the findings about the fractal parameters are revealing for future city planning and the spatial optimization of cities.     

Fractal dimension evolution and spatial replacement dynamics of urban growth

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 interpret 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 just 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 development. The 2-dimensional recurrence relations can be employed to generate the nonlinear dynamical behaviors such as bifurcation and chaos. A discovery is thus made in this article that, for the fractal dimension growth following the logistic curve, the normalized dimension value is the ratio of space filling. If the rate of spatial replacement (urban growth) is too high, the periodic oscillations and chaos will arise. The spatial replacement dynamics can be extended to general replacement dynamics, and bifurcation and chaos mirror a process of complex replacement.     

An analytical process of the spatio-temporal evolution of urban systems based on allometric and fractal ideas

This paper presents a computer-based analytical framework for the spatio-temporal evolution of urban systems using the ideas from the allometric growth associated with fractals. Both cities as systems and systems of cities follow the law of allometric growth, and the scaling factors of the allometric relations can compose the matrix equations as eigenfunctions. The fractal dimension arrays are just the eigenvectors of the scaling factor matrices while the numbers of variables are the greatest eigenvalues. The solutions of matrix equations can be employed to analyse city systems and evaluate relative levels of urban development. The method is applied to Hangzhou urban system of China. The results reveal clearly an urbanization process characterized as population concentration and an industrialization process characterized as industrial diffusion. The computation results are consistent with the reality, which indicate that the method is available for analyzing the spatio-temporal evolution of complex systems such as cities.     

Transportation systems and residential location

An optimization model for the spatial allocation of residents in an urban region is presented. The model allocates an exogenously given population on the basis of travel time (to workplaces) and population density criteria and subject to local and subregional capacity constraints. A way of calibrating the ‘value’ parameters of the model on a short-term plan proposal is outlined. Some applications of the model to planning problems of the Stockholm region are reported. These include sensitivity analyses of the population distribution of an adopted plan with respect to its exogenous assumptions (particularly those describing the transportation system). When evaluating investments with long life times, there is a strong need for long-term land use scenarios. Some long-range applications in the field of traffic and energy planning are reported. They can be seen as ‘prolongations’ of the adopted regional plan based on the preferences ‘revealed’ by the calibration procedure. Further extensions of the model to cover cost and energy considerations of urban developments are also outlined.     

Fractal systems of central places based on intermittency of space-filling

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 distribution of human settlements. This dimension value is inconsistent with empirical observations. Substituting the complete space filling with the incomplete space filling, we can obtain central place models with fractional dimension D < d = 2 indicative of spatial intermittency. Thus the conventional central place models are converted into fractal central place models. If we further integrate the chance factors into the improved central place fractals, the theory will be able to explain the real patterns of urban places very well. As empirical analyses, the US cities and towns are employed to verify the fractal-based models of central places.     

涟漪效应还是空间互动——基于空间面板联立方程模型的房价与地价关系研究

本文通过构建包含房价和地价在内的空间面板联立方程模型,利用2009~2016年中国286个地市的数据,构建空间面板联立方程模型,以研究房价和地价的空间自相关性和空间溢出效应。实证结果表明:房价与地价之间的空间互动作用明显,房价不但受到当地地价的影响还会受到周边城市房价的影响;而城市地价也会受到当地房价引起的引致需求影响,并且受到周边城市地价的影响。分区域来看,地价对房价的作用强度在东、中、西部地区依次下降,房价对地价作用强度呈现出相反的趋势。而从价格的溢出效应来看,无论是房价还是地价,西部地区的价格溢出效应都最为明显,表明相对于东、中部地区,西部地区城市在房价和地价上都更会受到周围城市的影响。     

Zipf’s law, 1/f noise, and fractal hierarchy

Fractals, 1/f noise, and Zipf’s laws are frequently observed within the natural living world as well as in social institutions, representing three signatures of complex systems. All these observations are associated with scaling laws and therefore have created much research interest in many diverse scientific circles. However, the inherent relationships between these scaling phenomena are not yet clear. In this paper, theoretical demonstration and mathematical experiments based on urban studies are employed to reveal the analogy between fractal patterns, 1/f spectra, and the Zipf distribution. First, the multifractal process empirically suggests the Zipf distribution. Second, a 1/f spectrum is mathematically identical to Zipf’s law. Third, both 1/f spectra and Zipf’s law can be converted into a self-similar hierarchy. Fourth, fractals, 1/f spectra, Zipf’s law can be rescaled with similar exponential laws and power laws. The self-similar hierarchy is a more general scaling method which can be used to unify different scaling phenomena and rules in both physical and social systems such as cities, rivers, earthquakes, fractals, 1/f noise, and rank-size distributions. The mathematical laws of this hierarchical structure can provide us with a holistic perspective of looking at complexity and complex systems.     

Mathematical models of urban growth

The article models the distribution of cities by population. Two approaches are considered to mathematical modeling of urban growth: a probability model in which the number of cities depends on the population and the rank model of distribution of cities by their population. Five population censuses are analyzed for Russia’s cities. The probability density function n(x, α) for the number of cities as a function of their population x is fitted to all the available censuses with a time-dependent coefficient α . The function α(t ) is approximated and a prediction for the nearest future is computed. In particular, it is shown that in 2010 compared with 2002 the number of large cities should increase, while the number of small town should decrease. A model is also proposed for the interaction of urban areas linked into a single hierarchical system. The model is based on a system of ordinary differential equations describing the change in urban population. Independently of the initial distribution, all the cities and town line up by the rank–size law and deviations from this law, as in real life, are observed only for some large and very small cities. Model parameters are fitted for Russia’s cities.     

中国城市土地利用效率的动态测度及影响因素——基于DEA-Tobit两步法的分析

利用数据包络模型(DEA)对中国大陆30个省、市、自治区2001—2010年城市土地利用效率进行了测度与评价,并运用Tobit回归模型对其影响因素进行了深入分析.结果表明:①中国城市土地利用效率总体呈现上升趋势,除少数省区效率值低于0.5以外,其余均保持在0.5-1之间;②城市土地利用效率前沿面分析结果显示,2001—2010年处于生产前沿面的省、市、自治区的数量保持稳定,但其空间分布却发生了一些变化,10年间东部地区减少2个,西部地区增加2个,中部地区一直处于零水平的状态;③城市土地利用过程中高投入低产出局面的扭转、非农从业人员效率的提高是城市土地利用效率改善的基本内核和关键所在;④经济规模对城市士地利用效率呈现正向促进作用,低层次的产业结构和不断增加的人口密度呈现负面影响,政府规制、空间集聚水平、城市规模和区位条件的影响并不显著.     

城市创新能力评价及时空格局演化研究

采用熵值法、基尼系数、探索性空间统计分析等方法从时间、空间两个维度对2007-2016年中国35个大中城市创新能力的评价、时空格局与演化规律进行了可视化分析。结果表明:(1)从时间维度看,大中城市创新能力在先期提升明显,末期有所回落,但仍呈现出整体上的提升;知识创新能力>政府支持与服务>创新基础环境>技术创新能力,技术短板突出;城市间创新能力的差异有向优化均衡方向发展的趋势,但创新基础环境的空间失衡加剧;G2> G1、G4> G3,城市之间创新能力的差异主要集中在技术创新能力上。(2)从空间维度看,莫兰检验显示样本期内城市创新能力存在显著的空间自相关性;与先期相比,2016年创新能力高水平城市数量增加,且表现出了更强的空间"集群化"特征,总体上呈现出"东高西低"的空间格局,具有明显的经济依赖性;样本期内大中城市创新能力的局部集聚模式变化不大,体现出一定的时空惯性,且高高集聚模式的城市具有较强的经济依赖特征;创新冷热点总体上呈"东热西冷"的空间分布格局。     

中国沿海城市人境旅游时空差异研究

改革开放以来,我国东部沿海城市入境旅游发展势头强劲,但沿海城市间的入境旅游发展却参差不齐.以53个沿海城市为区域研究单元,运用标准差、变异系数、基尼系数、泰尔指数等方法定量分析1995-2009年我国沿海城市入境旅游发展的时空差异过程及其特征,研究结论表明:我国沿海城市入境旅游绝对差异逐年扩大,相对差异缓慢缩小;入境旅游人数呈均衡化分散,旅游外汇收入存在局部区域集聚趋势;入境旅游泰尔指数具有三种典型演化特征,总体泰尔指数与区域内泰尔指数具有稳态渐进一致性,入境旅游集群特征明显,空间分布不平衡;沿海城市旅游发展水平与城市经济发展水平具有较高的地域分布一致性,入境旅游比国内旅游具有更大的内部差异,两者呈现空间非均衡分异特征.     

An allometric scaling law between gray matter and white matter of cerebral cortex

An allometric scaling relationship between cortical white and gray volumes is derived from a general model that describes brain’s remarkable efficiency and prodigious communications between brain areas. The model assumes that (1) a cell’s metabolic rate depends upon cell’s surface; (2) the overall basal metabolic rates of brain areas depend upon their fractal structures; (3) differential brain areas have same basal metabolic rate at slow wave sleep. The obtained allometric exponent scaling white matter to gray matter is 1.2, which is very much close to Zhang and Sejnowski’s observation data.     

湖南省城市生态竞争力比较研究

在现有城市生态竞争力分析的国内外研究成果的基础上,依据城市竞争力和生态系统相关理论,借鉴国内外评价机构的评价模型和分析体系,构建出基于PSR框架的湖南省地级城市生态竞争力的评价指标体系,并根据2012-2013年统计数据,运用直觉模糊综合评价模型,横向比较分析了湖南省14个地级市的城市生态竞争力情况,明确了现阶段影响湖南省各城市生态竞争力的主要因素,评价了各城市的现状及潜力以及各城市间所具有的竞争优势和劣势.在此基础上,有针对性地提出了促进湖南省各个城市生态竞争力提升的政策建议.     

A multidimensional approach to urban quality of life: The case of Istanbul

The purpose of this paper is to model the priorities, expectations and needs of the inhabitants of Istanbul, a city with a population of about 10 million, from a multidimensional perspective. In this way, effective allocation of the city’s resources can be achieved to improve the quality of life (QOL) for such a large number of people, which is the primary concern of the local authorities as well as the urban planners. For this reason, a survey is conducted in Istanbul so that the priorities of the inhabitants are revealed and the city where they would like to live is portrayed. The data obtained are used as input for hierarchical conjoint analysis. The survey is primarily based on the evaluation of hypothetical, experimentally designed city profiles for four different constructs on a 0–10 rating scale. The relative importance of the constructs is estimated through the eigenvector approach. The research is an interdisciplinary group work acting as a bridge between urban planning and multiattribute decision making; thus judgements of experts from different disciplines are used in every stage of the study.     

Scaling of mathematical fractals and box-counting quasi-measure

The same term, ‘fractals’ incorporates two rather different meanings and it is convenient to split the term into physical or empirical fractals and mathematical ones. The former term is used when one considers real world or numerically simulated objects exhibiting a particular kind of scaling that is the so-called fractal behaviour, in a bounded range of scales between upper and lower cutoffs. The latter term means sets having non-integer fractal dimensions. Mathematical fractals are often used as models for physical fractal objects. Scaling of mathematical fractals is considered using the Barenblatt–Borodich approach that refers physical quantities to a unit of the fractal measure of the set. To give a rigorous treatment of the fractal measure notion and to develop the approach, the concepts of upper and lower box-counting quasi-measures are presented. Scaling properties of the quasi-measures are studied. As examples of possible applications of the approach, scaling properties of the problems of fractal cracking and adsorption of various substances to fractal rough surfaces are discussed.     

Incorporating fractal concepts into equations of attrition for military conflicts

Using an agent-based model as a ‘battlefield laboratory’, we explore equations of combat attrition which extend beyond the conventional Lanchester equations and which endeavour to encapsulate the more complex aspects of warfare. Our approach compares predictions from candidate attrition equations with casualty data generated artificially from an agent-based model. For situations where the initial regimented structure of the fighting forces breaks down, introducing fractal concepts into the attrition equations proves effective at encapsulating complex aspects of the battle; with details in the time dependence of the casualty data able to be reproduced. Furthermore, measuring the fractal dimension of a fighting force's spatial distribution on the battlefield provides a sensitive probe of the combatants’ behaviour. Precise times at which key events occur during a battle can be pinpointed. This study furthers the body of work which considers warfare as a complex adaptive system and where fractal-like structures are expected to emerge.