Herding model and 1/f noise

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.

💡 Research Summary

The paper investigates how a simple agent‑based herding model can generate signals whose power spectral density follows a 1/f (or more generally 1/f^β) law. The authors start from the well‑known two‑state Kirman model, where each of N agents can be in state A (e.g., “chart”) or state B (“basic”). Transitions occur with a spontaneous rate ε and a herding term h that couples an agent’s decision to the number of agents already in the target state. Formally, the transition probabilities are

P(X→X+1) = (N−X)