A master equation for power laws

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.

Simultaneously low resistance drift and ultra‐fast phase change speed based on GeSb/Sb multilayer thin films

“…where Resistance drift is mainly caused by internal defects or degradation of the film. The power law equation is used to describe the resistance drift 34 :…”

“…Resistance drift is mainly caused by internal defects or degradation of the film. The power law equation is used to describe the resistance drift 34 : $$$$\begin{equation}{R}_t = {R}_0\ {\left(\frac{t}{{{\tau }_0}}\right)}^\alpha ,\ \end{equation}$$$$where $${R}_t$$ represents the measured resistance, R 0 and τ 0 are the initial constants, and α is the resistance drift coefficient measuring the drift speed. Figures 3A–C display the resistance drift of [GeSb(7 nm)/Sb(3 nm)] 5 , [GeSb(5 nm)/Sb(5 nm)] 5 , and pure Sb films at isothermal temperature 85°C, respectively.…”

Green Chemistry Allometry Test of Microwave-Assisted Synthesis of Transition Metal Nanostructures

“…From a statistical analysis point of view, a reliable verification of a power-law function is only considered when the fit is over 1.5 orders of magnitude or more, on both the x-axes and y-axes [18], Rau (2002) [22], Stumpf and Porter (2012) [23], Andriani and McKelvey (2019) [24], and Roman and Bertolotti (2022) [25]).…”

Global history, the emergence of chaos and inducing sustainability in networks of socio-ecological systems

“…Unfortunately, even an apparently simple system of low dimensionality can be complex enough to generate this kind of behavior [35]. Moreover, socio-environmental systems are subject to unforeseeable events, such as natural disasters [36], emergent social inequality and technological breakthroughs [37,38], which can introduce additional unpredictable features.…”