Modulated Branching Processes, Origins of Power Laws and Queueing Duality

Power law distributions have been repeatedly observed in a wide variety of socioeconomic, biological and technological areas. 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 protections may fail and companies 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 that are typically observed in proportional growth environments. Indeed, our main results show that the proposed mathematical models result in power law distributions under quite general polynomial Gartner-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.