Power-law distributions and Levy-stable intermittent fluctuations in stochastic systems of many autocatalytic elements

TL;DR

This paper models stochastic autocatalytic systems with many elements, revealing power-law distributions and Levy-stable fluctuations in their dynamics, with implications for understanding complex empirical systems.

Contribution

It introduces a coupled autocatalytic model showing power-law distributions and Levy-stable fluctuations, highlighting non-universal exponents and non-commuting limits.

Findings

Power-law distribution of $w_i$ with exponent $eta eq 1+ ext{alpha}$.

Intermittent fluctuations of the average follow a Levy-stable distribution.

Exponent $ ext{alpha}$ is insensitive to $ ext{Pi}( ext{lambda})$ but depends on $c$ and $N$.

Abstract

A generic model of stochastic autocatalytic dynamics with many degrees of freedom $w_i$ $i=1,...,N$ is studied using computer simulations. The time evolution of the $w_i$'s combines a random multiplicative dynamics $w_i(t+1) = \lambda w_i(t)$ at the individual level with a global coupling through a constraint which does not allow the $w_i$'s to fall below a lower cutoff given by $c \cdot \bar w$, where $\bar w$ is their momentary average and $0<c<1$ is a constant. The dynamic variables $w_i$ are found to exhibit a power-law distribution of the form $p(w) \sim w^{-1-\alpha}$. The exponent $\alpha (c,N)$ is quite insensitive to the distribution $\Pi(\lambda)$ of the random factor $\lambda$, but it is non-universal, and increases monotonically as a function of $c$. The "thermodynamic" limit, N goes to infty and the limit of decoupled free multiplicative random walks c goes to 0, do not commute: $\alpha(0,N) = 0$ for any finite $N$ while $ \alpha(c,\infty) \ge 1$ (which is the common range in empirical systems) for any positive $c$. The time evolution of ${\bar w (t)} $ exhibits intermittent fluctuations parametrized by a (truncated) L\'evy-stable distribution $L_{\alpha}(r)$ with the same index $\alpha$. This non-trivial relation between the distribution of the $w_i$'s at a given time and the temporal fluctuations of their average is examined and its relevance to empirical systems is discussed.