The rank-size distribution of settlements as a dynamic multifractal phenomenon

Haag, Günter

doi:10.1016/0960-0779(94)90063-9

Cited by 31 publications

(20 citation statements)

References 15 publications

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“…Multifractal geometry becomes a powerful tool for multi-scaling analysis of scale-free systems. It has been employed to characterize city-size distributions (Chen and Zhou, 2004;Haag, 1994), urban form and growth (Ariza-Villaverde et al, 2013;Chen and Wang, 2013;Frankhauser, 1998;Murcio et al, 2015), urban residential land price (Hu et al, 2012), regional population distributions (Appleby, 1996;Liu and Liu, 1993;Sé mé curbe et al, 2016), central place networks (Chen, 2014), bus-transport network (Pavón-Domí nguez et al, 2017), and so on. In the previous works, the multifractal method has been generalized for two times.…”

Section: Calculation Processmentioning

confidence: 99%

Spatial analysis of cities using Renyi entropy and fractal parameters

Chen¹,

Feng²

2017

Chaos, Solitons & Fractals

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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.

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Section: Calculation Processmentioning

confidence: 99%

Spatial analysis of cities using Renyi entropy and fractal parameters

Chen¹,

Feng²

2017

Chaos, Solitons & Fractals

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“…Recently, the multifractal dimension of rank-size distributions has been established (see Haag 1994;Mandelbrot 1997). For many urban analysts, however, the rank-size rule remains a quaint curiosity and their understanding of how a city self-organizes over time is modest at best.…”

Section: Arti®cial Citiesmentioning

confidence: 99%

Complex landscapes of spatial interaction

Batten¹

2001

The Annals of Regional Science

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How complex is the spatial economy? Some apostles of complexity argue that complex behaviour can arise in any system consisting of a largish number of intelligent, adaptive agents interacting on the basis of local information only. This paper examines several features of such dynamic systems, including path-dependence, emergence and self-organization. It goes on to explore their importance for the spatial sciences. Because space scales can change abruptly from local to global, strongly-interactive spatial economies sometimes exhibit astonishing collective properties, emergent features which are lawful in their own right. Segregation, self-similarity and the rank-size rule are familiar examples. To understand how collective order arises from seemingly random¯uctuations, we must note how agents choose to interact with other agents and with their environment. We must synthesize rather than analyse. In the paper, self-organization is explored in a variety of contexts, including Schelling's model of neighborhood segregation and some work with cellular automata that has sharpened our insights into the collective synthesis of agents' interactions. Power laws are widely observed. A new way of doing social science ± agent-based simulation ± o¨ers powerful new insights. It seems likely to revolutionize our ®eld, along with the whole of the social sciences. Some of the current research underway in this area is discussed.`T ruth is much too complicated to allow anything but approximations.'' john von neumann

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“…Cities and systems of cities in the real world are in fact multifractals with multi-scaling rather than monofractals with single scaling. Multifractal method has been applied to urban form (Ariza-Villaverde et al, 2013;Chen and Wang, 2013), regional population (Appleby, 1996;Chen and Shan, 1999), urban and rural settlement including central places and rank-size distributions, and so on (Chen, 2014b;Chen and Zhou, 2004;Haag, 1994;Hu et al, 2012 ;Liu and Chen, 2003).…”

Section: Introductionmentioning

confidence: 99%

Defining Urban and Rural Regions by Multifractal Spectrums of Urbanization

Chen

2016

Fractals

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The spatial pattern of urban-rural regional system is associated with the dynamic process of urbanization. How to characterize the urban-rural terrain using quantitative measurement is a difficult problem remaining to be solved. This paper is devoted to defining urban and rural regions using ideas from fractals. A basic postulate is that human geographical systems are of self-similar patterns associated with recursive processes. Then multifractal geometry can be employed to describe or define the urban and rural terrain with the level of urbanization. A spacefilling index of urban-rural region based on the generalized correlation dimension is presented to reflect the degree of geo-spatial utilization in terms of urbanization. The census data of America and China are adopted to show how to make empirical analyses of urban-rural multifractals. This work is not so much a positive analysis as a normative study, but it proposes a new way of investigating urban and rural regional systems using fractal theory.

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The rank-size distribution of settlements as a dynamic multifractal phenomenon

Cited by 31 publications

References 15 publications

Spatial analysis of cities using Renyi entropy and fractal parameters

Spatial analysis of cities using Renyi entropy and fractal parameters

Complex landscapes of spatial interaction

Defining Urban and Rural Regions by Multifractal Spectrums of Urbanization

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