We study the impact of settlement sizes on network connectivity in a spatial setting. First, we develop a model of geometric urban networks that posits a positive relationship between connectivity and size. Empirical evidence is then presented validating the model prediction that local links exhibit super-linear scaling with the exponent greater than 1, while long-range connections scale linearly with the unit exponent. The scaling exponents thus suggest that the impact of population size on connectivity is stronger within cities than between cities. We next combine the geometric framework with a computational model of interacting agents to generate a realistic settlement distribution and urban networks from the bottom-up. Calibrated simulation results demonstrate the consistency between hierarchical rank-size distribution and scale-free connectivity. Finally, coupling the spatial network with a tipping diffusion model allows us to consolidate the evolution of network connectivity, city sizes, and social practices in a unified computational framework. 
Size, connectivity, and tipping in spatial networks: Theory and empirics