Distance-weighted city growth

Diego Rybski, Anselmo García Cantú Ros, and Jürgen P. Kropp

Phys. Rev. E 87, 042114 – Published 16 April 2013

Abstract

Urban agglomerations exhibit complex emergent features of which Zipf's law, i.e., a power-law size distribution, and fractality may be regarded as the most prominent ones. We propose a simplistic model for the generation of citylike structures which is solely based on the assumption that growth is more likely to take place close to inhabited space. The model involves one parameter which is an exponent determining how strongly the attraction decays with the distance. In addition, the model is run iteratively so that existing clusters can grow (together) and new ones can emerge. The model is capable of reproducing the size distribution and the fractality of the boundary of the largest cluster. Although the power-law distribution depends on both, the imposed exponent and the iteration, the fractality seems to be independent of the former and only depends on the latter. Analyzing land-cover data, we estimate the parameter-value $γ \approx 2.5$ for Paris and its surroundings.

Received 17 September 2012

DOI:https://doi.org/10.1103/PhysRevE.87.042114

Authors & Affiliations

Diego Rybski^* and Anselmo García Cantú Ros

Potsdam Institute for Climate Impact Research–14412 Potsdam, Germany, EU

Jürgen P. Kropp

Potsdam Institute for Climate Impact Research–14412 Potsdam, Germany, EU and Institute of Earth and Environmental Science, University of Potsdam–14476 Potsdam, Germany, EU

^*ca-dr@rybski.de

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Vol. 87, Iss. 4 — April 2013

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Figure 1
Illustrative examples of model realizations for different iterations $i$ and different exponents $γ$ . (a)–(c) Different iterations of the model $i = 6, 10, 14 (γ = 2.5, N = 630)$ . Growth takes place close to occupied sites. (d)–(f) Realizations with different exponents $γ = 2.0, 2.5, 3.0$ (the occupation probability is $p \approx 0.04, N = 630$ ). The smaller $γ$ , the more scattered the emerging structures are, the larger $γ$ , the more compact they are.Reuse & Permissions
Figure 2
Cluster size distribution and dependence of the Zipf exponent on the occupation probability. (a) Examples of probability density distributions $P (S)$ of the cluster size $S$ disregarding the largest cluster of each realization: $i = 5, 10, 15$ (from top to bottom, $p \approx 0.004, 0.078, 0.573$ ), $γ = 2.5, N = 630$ , and $M = 100$ (solid lines connect symbols). The two green dotted squares represent the contribution of the largest cluster ( $i = 10$ ). (b) The obtained Zipf exponents $ζ$ as a function of the occupation probability $p$ : $γ = 2, 2.5, 3, 3.5$ (from top to bottom), $N = 630, M = 100$ . Solid lines are fits according to Eq. (3). Dashed lines indicate $ζ = 2$ .Reuse & Permissions
Figure 3
Parameter values $a, b$ , and $c$ as a function of the exponent $γ$ from fitting the Zipf exponent versus the occupation probability [Fig. 2b]. The dots represent the parameters obtained from fitting Eq. (3) to the numerical values, the solid lines are given by $a = - k_{1} γ + k_{2}, b_{γ \leq 3} = - e^{- k_{3} γ + k_{4}}$ , $b_{γ > 3} = - e^{k_{3} γ - k_{5}}$ , and $c = e^{- k_{6} γ + k_{7}} - e^{- k_{8} + k_{7}}$ ( $k_{1} - k_{8}$ are fitting parameters). In order to have enough values, we do not separate the different system sizes.Reuse & Permissions
Figure 4
Self-similar scaling of the boundary of the largest cluster. (a) Number of boxes necessary to cover the boundary as a function of the size of the boxes: $γ = 2.5, N = 630, i = 5, 10, 15 (p \approx 0.004, 0.078, 0.573)$ , and $M = 100$ . The results follow Eq. (4) indicating the fractal property of the boundary with a fractal dimension $1 < d_{B} < 2$ . The error bars represent standard deviations among the realizations. (b) Fractal dimension as a function of the occupation probability for $γ = 2.0, 2.5, 3.0, 3.5 (N = 630$ and $M = 100)$ . $d_{B}$ increases approximately logarithmically, independent of $γ$ .Reuse & Permissions
Figure 5
Percolation transition of the model. (a) Average size of the finite clusters: $γ = 3.8, 3.6, 3.4, 3.2, 3.0$ (from top to bottom), $N = 210$ , and $M = 1000$ . The vertical line represents the percolation transition of uncorrelated percolation $p_{c}^{*} ≃ 0.593$ (site percolation in the square lattice, [36]). The maximum is located at the percolation transition. (b) Percolation transition $p_{c}$ as a function of $γ : N = 210, 420, 630 (M = 1000, 400, 100)$ . For small $γ$ , the percolation transition is close to $p_{c}^{*}$ as indicated by the horizontal line.Reuse & Permissions
Figure 6
Urban growth and the estimation of $γ$ for Paris and its surroundings in the years 2000 and 2006. (a) Considered land-cover data in a window of $1000 \times 1000$ grid points (250 m resolution). The panel distinguishes the following sites: dark red: populated in 2000; gray: unpopulated in 2000; blue: water; light red: $nonurban \to urban$ ; and yellow: $urban \to nonurban$ . (b) Logarithmic likelihood of urbanization. First, we calculate the probabilities $q_{j}$ according to Eq. (1) for the year 2000, whereas ( $w_{k} = 1$ for urban and $w_{k} = 0$ for nonurban cells). Then, we determine $Q$ according to Eq. (5) over all nonurban cells in 2000 (the change urban to nonurban is disregarded). We find a maximum at $γ_{Paris} \approx 2.5$ .Reuse & Permissions

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