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A master equation for power laws.

Sabin Roman1,2, Francesco Bertolotti2

  • 1Centre for the Study of Existential Risk, University of Cambridge, Cambridge, UK.

Royal Society Open Science
|December 9, 2022
PubMed
Summary
This summary is machine-generated.

We introduce a novel mechanism for generating power laws from random walks and Markov chains. This method explains cascade size distributions and offers a natural cut-off, aligning with empirical data like Richardson's Law.

Keywords:
Markov chainspower lawsurn model

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Area of Science:

  • Statistical Physics
  • Complex Systems
  • Mathematical Modeling

Background:

  • Power laws are ubiquitous in natural and social phenomena.
  • Existing models often lack a clear mechanistic derivation or natural cut-off.
  • Understanding the generation of power-law distributions is crucial for diverse scientific fields.

Purpose of the Study:

  • To propose a new, mechanism-based approach for generating power-law distributions.
  • To derive a master equation describing the temporal evolution of cascade sizes.
  • To provide a theoretical justification for power-law exponents and natural cut-offs.

Main Methods:

  • Derivation of the Fokker-Planck equation from a random walk model.
  • Development of a master equation for power laws based on a Markov chain.
  • Solving the partial differential equation to obtain a closed-form solution for cascade size distributions.
  • Introduction of an urn model to further illustrate power-law generation.

Main Results:

  • A closed-form solution for the master equation, explicitly linking cascade number, size, and time.
  • A natural cut-off in the power-law solution due to finite time horizons.
  • Theoretical justification for a power-law exponent of 2, consistent with empirical data.
  • Demonstration of power-law behavior across all cascade sizes via log-log plots.

Conclusions:

  • The proposed mechanism provides a robust framework for generating power laws with natural cut-offs.
  • The derived master equation and its solutions offer valuable insights into the dynamics of cascading processes.
  • The model's ability to explain empirical distributions like Richardson's Law highlights its broad applicability.