Urban settlement transitions
Urban growth dynamics attracts the efforts of scientists from many different disciplines with objectives ranging from theoretical understanding to the development of carefully tuned realistic models that can serve as planning and policy tools. Theoretical models are often abstract and of limited applied value while most applied models yield little theoretical understanding. Here we present a mathematically well-defined model based on a modified Markov random field with lattice-wide interactions that produces realistic growth patterns as well as behavior observed in a range of other models based on diffusion-limited aggregation, cellular automata, and similar models. We investigate the framework's ability to generate plausible patterns using minimal assumptions about the interaction parameters since the tuning and specific definition of these are outside of the scope of this paper. Typical universality classes of the simulated dynamics and the phase transitions between them are discussed in the context of real urban dynamics. Using suitability data derived from topography, we produce configurations quantitatively similar to real cities. Also, an intuitive class of interaction rules is found to produce fractal configurations, not unlike vascular systems, that resemble urban sprawl. The dynamics are driven by interactions, depicting human decisions, between all lattice points. This is realized in a computationally efficient way using a mean-field renormalization (area average) approach. The model provides a mathematically transparent framework to which any level of detail necessary for actual urban planning application can be added.