Modulated Branching Processes and Origins of Power Laws

摘要

Power law distributions have been repeatedly observed in a wide variety of socioeconomic, biological and technological areas, including: distributions of wealth, species-area relationships, populations of cities, values of companies, sizes of living organisms and, more recently, distributions of documents and visitors on the Web, etc. In the vast majority of these observations, e.g., city populations and sizes of living organisms, the objects of interest evolve due to the replication of their many independent components, e.g., births-deaths of individuals and replications of cells. Furthermore, the rates of replication of the many components are often controlled by exogenous parameters causing periods of expansion and contraction, e.g., baby booms and busts, economic booms and recessions, etc In addition, the sizes of these objects often either have reflective lower boundaries, e.g., cities do not fall bellow a certain size, low income individuals are subsidized by the government, companies are protected by bankruptcy laws, etc; or have porous/absorbing lower boundaries, e.g., cities may degenerate, bankruptcy protection may fail and a company can be liquidated. Hence, it is natural to propose reflected modulated branching processes as generic models for many of the preceding observations of power laws. Indeed, our main results show that these apparently new mathematical objects result in power law distributions under quite general "polynomial Gaertner-Ellis" conditions. The generality of our results could explain the ubiquitous nature of power law distributions. Furthermore, an informal interpretation of our main results suggests that alternating periods of expansion and reduction, e.g., economic booms and recessions, are primarily responsible for the appearance of power law distributions. Our results also establish a general asymptotic equivalence between the reflected branching processes and the corresponding reflected multiplicative processes. Furthermore, in the course of our analysis, we discover a duality between the reflected multiplicative processes and queueing theory. Essentially, this duality demonstrates that the power law distributions play an equivalent role for reflected multiplicative processes as the exponential/geometric distributions do in queueing analysis.