one of the oldest and most popular models proposes that population density shows a simple exponential decay with radial distance from he city centre: \rho ® = \rho_0 exp ^{-br} – C. Clark “Urban Population Densities”, Journal of the Royal Statistical Society vol 114 no. 4 pp.490-496 (1951).
This form is analytically tractable and shows good agreement in many empirically studied cases –by J. C. Martori and J. Surinach, “Classical Models of urban population density”, Congress of the European Regional Science Association, 2001
Newling suggests modifying this to a quadratic exponential function \rho® = \rho_0 exp^{\br-cr^2} in order to add the flexibility to account for the density crater seen towards the centre of some larger cities [B. E. Newling, “The Spatial Variation of Urban population Densities”, Geographical Review 59 2 242-252 (1969).
Decaying power-law function: \rho® = K r^{-a}: First introduced by Smeed. although there is some analytical awkwardness it has some empirical advantage. by R. J. Smeed, The traffic Problem in Towns (Manchester Statistical Society Papers). Norbury Lockwood: Manchester, 1961.
Combined power-law with gamma function: \rho® = K r^{-ar} exp^{-br}: S. Angel and G. M. Hyman, Urban fields: a geometry of movement for regional science. Pion: London (1976).
common characteristics of these models is radial symmetry
Batty and Kim (1992) develop an interesting argument in favor of using a power-law by M. Batty and K. S. Kim “form Follows Function: Reforumulating Urban population density functions”, Urban studies vol. 29, no. 7 pp. 1043-1069 (1992). They find \alpha, the exponent of the fractal structure of cities. Cities tend to exhibit substantial self-similarity on different spatial scales and often have distinct hierarhical structures– by M. Batty and P. A. Longley, Fractal Cities: A geometry of Form and Function. Academic Press: London 1994; and by S. Guoqiang, “Fractal dimension and fractal growth of urbanized areas”, Int. J. Geographical Information Science vol. 16, no. 5, pp.419-437 (2002). and by L. Yongmei and T. Junmei, “Fractal dimension of a transportation network and its relationship with urban growth: a study of the Dallas-Fort Worth area” Env. and Planning B: Planning and Design vol. 31 pp. 895-911 (2004).
Simulated cities “grown” with a diffusion-limited aggregation (DLA) method were found to behave similarly - an encouraging correspondence between this fractal interpretation and a popular statistical model by M. Batty and P. Longley, “Urban growth and form: scaling, fractal geometry, and diffusion-limited aggregation”, Environment and Planing A. vol 21 no. 11 pp.1447-1472 (1989).
Multi-center distribution of London (barthelemey)