Abstract
The origin(s) of the ubiquity of probability distribution functions with power law tails is still a matter of fascination and investigation in many scientific fields from linguistic, social, economic, computer sciences to essentially all natural sciences. In parallel, self-excited dynamics is a prevalent characteristic of many systems, from the physics of shot noise and intermittent processes, to seismicity, financial and social systems. Motivated by activation processes of the Arrhenius form, we bring the two threads together by introducing a general class of nonlinear self-excited point processes with fast-accelerating intensities as a function of "tension." Solving the corresponding master equations, we find that a wide class of such nonlinear Hawkes processes have the probability distribution functions of their intensities described by a power law on the condition that (i)the intensity is a fast-accelerating function of tension, (ii)the distribution of marks is two sided with nonpositive mean, and (iii)it has fast-decaying tails. In particular, Zipf's scaling is obtained in the limit where the average mark is vanishing. This unearths a novel mechanism for power laws including Zipf's law, providing a new understanding of their ubiquity.
Highlights
Faculty of Engineering, Information and Systems, University of Tsukuba, Tennodai, Tsukuba, Ibaraki 305-8573, Japan and JST, PRESTO, 4-1-8 Honcho, Kawaguchi, Saitama 332-0012, Japan
Self-excited dynamics is a prevalent characteristic of many systems, from the physics of shot noise and intermittent processes, to seismicity, financial and social systems
Motivated by activation processes of the Arrhenius form, we bring the two threads together by introducing a general class of nonlinear self-excited point processes with fast-accelerating intensities as a function of “tension.” Solving the corresponding master equations, we find that a wide class of such nonlinear Hawkes processes have the probability distribution functions of their intensities described by a power law on the condition that (i) the intensity is a fast-accelerating function of tension, (ii) the distribution of marks is two sided with nonpositive mean, and (iii) it has fast-decaying tails
Summary
Faculty of Engineering, Information and Systems, University of Tsukuba, Tennodai, Tsukuba, Ibaraki 305-8573, Japan and JST, PRESTO, 4-1-8 Honcho, Kawaguchi, Saitama 332-0012, Japan. We study a class of nonlinear Hawkes processes characterized by fast-accelerating intensities as a function of an auxiliary field called the “tension,” and report the first explicit solution that is applicable to a wide class of nonlinear Hawkes processes. We find that this class of nonlinear Hawkes family universally exhibits intensity distributions with power law tails.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
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 call
