Scaling laws and indications of self-organized criticality in urban systems

Evolution of urban systems has been considered to exhibit some form of self-organized criticality (SOC) in the literature. This paper provides further mathematical foundations and empirical evidences to support the supposition. The hierarchical structure of systems of cities can be formulated as three exponential functions: the number law, the population size law, and the area law. These laws are identical in form to the Horton–Strahler laws of rivers and Gutenberg–Richter laws of earthquakes. From the exponential functions, three indications of SOC are also derived: the frequency–spectrum relation indicting the 1/f noise, the power laws indicating the fractal structure, and the Zipf’s law indicating the rank-size distribution. These mathematical models form a set of scaling laws for urban systems, as demonstrated in the empirical study of the system of cities in China. The fact that the scaling laws of urban systems bear an analogy to those on rivers and earthquakes lends further support to the notion of possible SOC in urban systems.