Distance weighted city growth

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 city-like 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. While 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 $\gamma\approx 2.5$ for Paris and it's surroundings.