Modelling urban growth patterns

Cities grow in a way that might be expected to resemble the growth of two-dimensional aggregates of particles, and this has led to recent attempts^1–3 to model urban growth using ideas from the statistical physics of clusters. In particular, the model of diffusion-limited aggregation^4,5 (DLA) has been invoked to rationalize the apparently fractal nature of urban morphologies¹. The DLA model predicts that there should exist only one large fractal cluster, which is almost perfectly screened from incoming ‘development units’ (representing, for example, people, capital or resources), so that almost all of the cluster growth takes place at the tips of the cluster’s branches. Here we show that an alternative model, in which development units are correlated rather than being added to the cluster at random, is better able to reproduce the observed morphology of cities and the area distribution of sub-clusters (‘towns’) in an urban system, and can also describe urban growth dynamics. Our physical model, which corresponds to the correlated percolation model^6–8 in the presence of a density gradient⁹, is motivated by the fact that in urban areas development attracts further development. The model offers the possibility of predicting the global properties (such as scaling behaviour) of urban morphologies.