The Lévy Flight of Cities: Analyzing Social-Economical Trajectories with Auto-Embedding

Lévy flights are useful in stochastic measurement, such as biology, human mobility, and financial mathematics, where trajectories contain many short flights and some long flights, and the step-lengths follow a power-law distribution. For cities inhabited by human beings, this paper proposes to use latent factors to study social-economical development trajectories and observes that the trajectories of four Asian cities exhibit Lévy flight characteristics. We collect the social and economical data such as GDP, goods producing industries, population, and general tertiary education, and map these data into social or economical factor through a deep-learning embedding method auto-encoder. We find that the step-lengths of these urban social-economical trajectories can be fitted approximately as a power-law distribution. We use the stochastic multiplicative processes (SMPs) to explain such pattern, wherein the presence of a boundary constraint, the SMP, leads to a power-law distribution. It means that these urban social-economical trajectories follow a Lévy flight pattern, where some years have large changes and many years have little changes.