A Class of Markovian Self-Reinforcing Processes with Power-Law Distributions

TL;DR

This paper introduces a new Markovian self-reinforcing process that models bursty natural and social phenomena with power-law distributions, offering a simpler and more tractable alternative to existing non-local models.

Contribution

The authors develop a local, Markovian nonlinear process that captures power-law bursty dynamics, improving mechanistic understanding and analytical tractability over prior non-local models.

Findings

Generates power-law tails for inter-event intervals and event rates

More analytically tractable and easier to simulate than Hawkes processes

Applicable to a broad range of natural and social bursty phenomena

Abstract

Solar flares, email exchanges, and many natural or social systems exhibit bursty dynamics, with periods of intense activity separated by long inactivity. These patterns often follow power- law distributions in inter-event intervals or event rates. Existing models typically capture only one of these features and rely on non-local memory, which complicates analysis and mechanistic interpretation. We introduce a novel self-reinforcing point process whose event rates are governed by local, Markovian nonlinear dynamics and post-event resets. The model generates power-law tails for both inter-event intervals and event rates over a broad range of exponents observed empirically across natural and human phenomena. Compared to non-local models such as Hawkes processes, our approach is mechanistically simpler, highly analytically tractable, and also easier to simulate. We provide methods for model fitting and validation, establishing this framework as a versatile foundation for the study of bursty phenomena.