Markov Chain Hidden behind Power Law Mechanism of Self-organized Criticality

TL;DR

This paper introduces a four-state Markov model to analytically derive the power law distribution in Self-Organized Criticality systems, offering a new theoretical perspective on the universality of power laws.

Contribution

It presents a novel Markov model and a mathematical proof that explain the power law distribution in SOC, expanding understanding beyond simulation-based approaches.

Findings

Derived a mathematical proof of power law distribution in SOC

Showed the universality of power laws in a broader class of dynamical systems

Provided a new analytical framework for SOC analysis

Abstract

To describe and analyze the dynamics of Self-Organized Criticality (SOC) systems, a four-state continuous-time Markov model is proposed in this paper. Different to computer simulation or numeric experimental approaches commonly employed for explaining the power law in SOC, in this paper, based on this Markov model, using E.T.Jayness' Maximum Entropy method, we have derived a mathematical proof on the power law distribution for the size of these events. Both this Makov model and the mathematical proof on power law present a new angle on the universality of power law distributions, they also show that the scale free property exists not necessary only in SOC system, but in a class of dynamical systems which can be modelled by the proposed Markov model.