Chapter 27 Comparing power laws and exponentials in simulations of gravitational growth
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Urban gravitation represents an established process of city morphology, based on two "axioms", i.e. the attraction force of a center (agglomeration externalities) and the friction of space due to geographical separation. However, there is no consensus about the actual form of the attraction function. Here we compare power-law and exponential attraction in the stochastic gravitation model. Therefore, we run numerical simulations with both forms and analyze the generated structures. Specifically, we compare the area-perimeter relation, the perimeter width, the cluster size distribution, and the probability as a function of the distance to the closest urban cell. The area-perimeter relation indicates that the perimeter of the central cluster generated by the exponential attraction approximately corresponds to a circle. The width of the perimeters generated by the power-law attraction is larger than the one of those generated by the exponential attraction. According to the simulations, exponential attraction leads to somewhat large Zipf-exponents -- this means a rather uneven number of small and large cities emerges. For exponential attraction, the likelihood of urban development vanishes at short distances from the largest cluster, which inhibits the emergence of satellite clusters. In summary, power-law attraction generates more realistic patterns than exponential attraction. We also observe that the model-parameter _ of the power-law attraction mostly influences the properties of the smaller surrounding clusters while the characteristics of the main cluster are rather constant.

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