A Master Equation for Power Laws

TL;DR

This paper introduces a master equation approach to generate power laws, deriving a new model from Markov chains that explains cascade behaviors with natural cut-offs and matches empirical data like conflict sizes.

Contribution

It presents a novel master equation for power laws derived from Markov chains, providing explicit solutions with natural cut-offs and empirical validation.

Findings

Derives a master equation with a closed-form solution for power laws.

Explains the natural cut-off in power law distributions due to finite cascade sizes.

Matches empirical data such as conflict size distributions with an exponent of 2.

Abstract

We propose a new mechanism for generating power laws. Starting from a random walk, we first outline a simple derivation of the Fokker-Planck equation. By analogy, starting from a certain Markov chain, we derive a master equation for power laws that describes how the number of cascades changes over time (cascades are consecutive transitions that end when the initial state is reached). The partial differential equation has a closed form solution which gives an explicit dependence of the number of cascades on their size and on time. Furthermore, the power law solution has a natural cut-off, a feature often seen in empirical data. This is due to the finite size a cascade can have in a finite time horizon. The derivation of the equation provides a justification for an exponent equal to 2, which agrees well with several empirical distributions, including Richardson's law on the size and frequency of deadly conflicts. Nevertheless, the equation can be solved for any exponent value. In addition, we propose an urn model where the number of consecutive ball extractions follows a power law. In all cases, the power law is manifest over the entire range of cascade sizes, as shown through log-log plots in the frequency and rank distributions.