Research On Urban Power Law Distribution And Urban Allometric Scaling

Urbanization is an important process in human society.On the one hand,urban-ization can bright us high income,better medical,health and education services.On the other hand,urbanization can also bright us traffic congestion,environmental pol-lution,rising crime rates,opportunities for inequality,etc.Urbanization has brought us unprecedented challenges while citis are the crucible of urbanization.Therefore,it s imperative to understand the dynamic,growth and evolution in a predictable and quantitative way.In the research of science of cities,the power law distribution and urban scaling are both the landmark of study results.In theory,they can help us for bet-ter understanding urban structure,growth and evolution.In practical,they can supply suggestions for urban planning and urban sustainable development.In this thesis,urban density was taken as an example to build the power law dis-tribution inside cities.We first transformed the discrete data to continuous data,then equidistanting and grouping them.Then Nonlinear Least Squares Estimator(OLS)and Levenberg-Marquardt Method(LM)was used to fit the population density and area data.We tested the goodness fit of the distribuion by adjusted determined coefficient R2,Akaike Information Criterion(AIC)and B ayesian Information Criterion(BIC),and tested the significance of the model by F test.Then we explain the connotation of the power law distribution in detail.After this,the dynamic of the power law distribution inside cities can be obtained.Then we model the dynamic of this power law distribution.At last,the connection between power law distribution and urban allometric scaling was built by simulation based on the power law distribution and its dynamics.Then can explain the origins of the urban scaling between area and population.This method can effectively avoid the existed gap between hypothesis and reality proposed by previous models.The research of the mechnical of urban scaling between area and population can also provide reference for the allmetric scaling in biology.The main conclusions area as follows:(1)The power-law distribution between population density and area fills the gap in urban distribution within the city.This distribution belongs to the three-parameter power function form,which is the Bleasdale-Nelder distribution(y=(a+bx)-1/c)used in this paper,which effectively avoids the power law distribution in the city size distribution(A breakpoint problem with two parameter power law distributions,namely y=axb).The three-parameter Bleasdale-Nelder power law distribution fully illustrates the degree of differentiation between the various components within the urban system and characterizes the heterogeneity of the internal structure of the urban system.(2)The dynamic change of the power law distribution of population density and area belongs to the horizontal change,which makes up for the deficiency of the dynamic change of urban power law distribution in the horizontal direction.As urban area in-creases,both the original data,the fitting curve,and the model curve have a clear upward trend.The parameters of the Bleasdale-Nelder distribution model,a,b,c,and the dis-tribution of the area are in the form of power law.The parameter of a tends to decrease with the increase of urban area.The b value and the c value tend to decrease as the urban area increases.The linear distribution coefficient A and the area distribution are in the form of a quadratic polynomial distribution.The linear distribution coefficient B and the area distribution are in the form of a cubic polynomial distribution.From the dy-namic change of the population density boundary,the minimum and maximum values of the boundary remain constant with the increase of the area.The maximum population density and urban area are consistent with the quadratic polynomial distribution.From the perspective of change trends,the maximum population density shows an increasing trend with the increase of urban area.The dynamic changes in the Gibrat simulation show that,at any stage of urban development,the rate of change of area increases first and then increases with the increase of the initial area,which is consistent with the con-clusion reached by Gibrat.The dynamic change of Simon s simulation shows that no matter at which stage of urban development,the dynamic rate of change of area mean increases rapidly with the increase of initial area,and this conclusion is consistent with Simon s conclusion.(3)The distribution of population density and area determines the urban allometric scaling between population and area.(4)Changes in the distribution parameters of different population densities and ar-eas will lead to changes in population and area urban allometric scaling coefficients and exponent.Specifically,the heterogeneity distribution of population density and area,the heterogeneity distribution of population density and population,and the systematic change of the maximum density with urban area lead to the allometric scaling between population and area.