Herding model and 1/f noise

TL;DR

This paper demonstrates that a simple herding agent-based model can produce signals with 1/f noise, linking microscopic herding behavior to macroscopic power-law spectral properties.

Contribution

It derives a non-linear stochastic differential equation from herding dynamics that explains the emergence of 1/f^beta noise, providing a microscopic basis for such signals.

Findings

Herding behavior can generate 1/f spectral density in signals.

Non-linear stochastic differential equations model 1/f^beta noise.

Feedback mechanisms enhance power-law statistics in the model.

Abstract

We provide evidence that for some values of the parameters a simple agent based model, describing herding behavior, yields signals with 1/f power spectral density. We derive a non-linear stochastic differential equation for the ratio of number of agents and show, that it has the form proposed earlier for modeling of 1/f^beta noise with different exponents beta. The non-linear terms in the transition probabilities, quantifying the herding behavior, are crucial to the appearance of 1/f noise. Thus, the herding dynamics can be seen as a microscopic explanation of the proposed non-linear stochastic differential equations generating signals with 1/f^beta spectrum. We also consider the possible feedback of macroscopic state on microscopic transition probabilities strengthening the non-linearity of equations and providing more opportunities in the modeling of processes exhibiting power-law statistics.