Abstract
Many phenomena with power laws have been observed in various fields of the natural and social sciences, and these power laws are often interpreted as the macro behaviors of systems that consist of micro units. In this paper, we review some basic mathematical mechanisms that are known to generate power laws. In particular, we focus on stochastic processes including the Yule process and the Simon process as well as some recent models. The main purpose of this paper is to explain the mathematical details of their mechanisms in a self-contained manner.
Highlights
Many phenomena with power laws have been observed in various fields of the natural and social sciences: physics, biology, earth planetary science, computer science, economics, and so on
We focus on the generating mechanisms with the stochastic processes in the above list4: the growth and preferential attachment and the stochastic models based on the Geometric Brownian motion (GBM), which, in particular, are widely applied in social science
We have reviewed nine generating mathematical mechanisms of power laws (i.e., Yule process, Simon process, Barabási– Albert model, geometric Brownian motion with a reflecting wall and reset events, Kesten process, Generalized Lotka–Volterra model, and Bouchaud–Mézard model, and the combination of exponentials) that are widely applied in the social sciences
Summary
Many phenomena with power laws have been observed in various fields of the natural and social sciences: physics, biology, earth planetary science, computer science, economics, and so on These power laws can be interpreted as the macro behaviors of the systems that consist of micro units (i.e., agents, individuals, particles, and so on). We focus on the generating mechanisms with the stochastic processes in the above list: the growth and preferential attachment and the stochastic models based on the GBM, which, in particular, are widely applied in social science. We mainly give full details of the mathematical formalisms for these mechanisms in selfcontained manner, because understanding them is important for researchers in any field to create new models generating power laws in empirical data. The necessary mathematical supplements to understand these mechanisms are given in the Appendix at the end of this paper
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
