Fluctuation scaling in complex systems: Taylor's law and beyond

TL;DR

This paper reviews the widespread occurrence of fluctuation scaling, known as Taylor's law, across various complex systems, and introduces new empirical data and theoretical insights into its underlying principles and models.

Contribution

It provides a comprehensive survey of fluctuation scaling, reports new empirical data, and proposes a mean-field theoretical framework explaining the phenomenon's universality.

Findings

Fluctuation scaling is observed across diverse complex systems.

A mean-field model explains the emergence of fluctuation scaling.

Limit theorems underpin the generality of Taylor's law.

Abstract

Complex systems consist of many interacting elements which participate in some dynamical process. The activity of various elements is often different and the fluctuation in the activity of an element grows monotonically with the average activity. This relationship is often of the form "$fluctuations \approx const.\times average^\alpha$", where the exponent $\alpha$ is predominantly in the range $[1/2, 1]$. This power law has been observed in a very wide range of disciplines, ranging from population dynamics through the Internet to the stock market and it is often treated under the names \emph{Taylor's law} or \emph{fluctuation scaling}. This review attempts to show how general the above scaling relationship is by surveying the literature, as well as by reporting some new empirical data and model calculations. We also show some basic principles that can underlie the generality of the phenomenon. This is followed by a mean-field framework based on sums of random variables. In this context the emergence of fluctuation scaling is equivalent to some corresponding limit theorems. In certain physical systems fluctuation scaling can be related to finite size scaling.