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Form Follows Function: Reformulating Urban Population Density Functions

Author

Listed:
  • Michael Batty

    (National Center for Geographic Information and Analysis, State University of New York at Buffalo, 301F Wilkeson Quad, Buffalo, NY 14261, USA)

  • Kwang Sik Kim

    (Department of Public Administration, Sung Kyun Kwan University, 53, 3-KA, Myungryun-dong, Chongro-ku, Seoul 110-745, Korea)

Abstract

In this paper, we argue that the most appropriate form for urban population density models is the inverse power function, contrary to conventional practice, which is largely based upon the negative exponential. We first show that the inverse power function has several theoretical properties which have hitherto gone unremarked in the literature. Our main argument, however, is based on the notion that a density function should describe the extent to which the space available for urban development is filled. To this end, we introduce ideas from urban allometry and fractal geometry to demonstrate that the inverse power model is the only function which embodies the fractal property of self-similarity which we consider to be a basic characteristic of urban form and density. In short, we show that the distance parameter a of the inverse power model is a measure of the extent to which space is filled, and that its value is determined by the basic relation D +α=2 where D is the fractal dimension of the city in question. We then test this model using four data sets which measure the density and morphology of the city of Seoul. Using a variety of estimation methods such as loglinear regression, dimensional approximation and entropy-maximising, we estimate dimension and density parameters for 136 variants of the function and its data sets. From these estimates, 125 are within the values hypothesised and this suggests fairly conclusively that the value of the density parameter a for the inverse power function should be within the range 0-1 and probably between 0.2 and 0.5. Many related questions are raised by this analysis which will form the subject of future research.

Suggested Citation

  • Michael Batty & Kwang Sik Kim, 1992. "Form Follows Function: Reformulating Urban Population Density Functions," Urban Studies, Urban Studies Journal Limited, vol. 29(7), pages 1043-1069, October.
  • Handle: RePEc:sae:urbstu:v:29:y:1992:i:7:p:1043-1069
    DOI: 10.1080/00420989220081041
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    References listed on IDEAS

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    1. Herbert K. Weiss, 1961. "The Distribution of Urban Population and an Application to a Servicing Problem," Operations Research, INFORMS, vol. 9(6), pages 860-874, December.
    2. R.H. Best & A.R. Jones & A.W. Rogers, 1974. "The Density-size Rule," Urban Studies, Urban Studies Journal Limited, vol. 11(2), pages 201-208, June.
    3. A. Stewart Fotheringham & Michael Batty & Paul A. Longley, 1989. "Diffusion‐Limited Aggregation And The Fractal Nature Of Urban Growth," Papers in Regional Science, Wiley Blackwell, vol. 67(1), pages 55-69, January.
    4. Beckmann, Martin J., 1969. "On the distribution of urban rent and residential density," Journal of Economic Theory, Elsevier, vol. 1(1), pages 60-67, June.
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    Cited by:

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    2. Marjo Kasanko & Jose I. Barredo & Carlo Lavalle & Valentina Sagris, 2005. "Towards urban un-sustainability in Europe? An indicator-based analysis," ERSA conference papers ersa05p149, European Regional Science Association.
    3. Daniel Griffith & David Wong, 2007. "Modeling population density across major US cities: a polycentric spatial regression approach," Journal of Geographical Systems, Springer, vol. 9(1), pages 53-75, April.
    4. Luca Salvati, 2019. "Examining urban functions along a metropolitan gradient: a geographically weighted regression tells you more," Letters in Spatial and Resource Sciences, Springer, vol. 12(1), pages 19-40, April.
    5. António Dinis F. Santos & Duarte Valério & J. A. Tenreiro Machado & António M. Lopes, 2019. "A fractional perspective to the modelling of Lisbon’s public transportation network," Transportation, Springer, vol. 46(5), pages 1893-1913, October.
    6. Mario Coccia & Barry Bozeman, 2016. "Allometric models to measure and analyze the evolution of international research collaboration," Scientometrics, Springer;Akadémiai Kiadó, vol. 108(3), pages 1065-1084, September.
    7. Jaume Benseny & Juuso Toyli & Heikki Hammainen & Andr'es Arcia-Moret, 2019. "The mitigating role of regulation on the concentric patterns of broadband diffusion. The case of Finland," Papers 1905.03002, arXiv.org, revised Oct 2020.
    8. Ondřej Slach & Vojtěch Bosák & Luděk Krtička & Alexandr Nováček & Petr Rumpel, 2019. "Urban Shrinkage and Sustainability: Assessing the Nexus between Population Density, Urban Structures and Urban Sustainability," Sustainability, MDPI, vol. 11(15), pages 1-22, August.

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