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.
Introduction. -Kirman's seminal herding model was introduced in Refs. [1,2]. This is a simple stochastic model of information transmission initially designed to explain the herding behavior in ant colonies, gathering food from two identical sources located in their neighborhood. Kirman noticed that entomologists and economists observe similar patterns in rather different systems. If there are two identical food sources available to ants in a colony, majority of ants still tend to use only a single food source at any given time. Furthermore, switches to the new food source occur despite the fact that food sources remain identical [3]. Human crowd behavior tends to be quite similar, at least in statistical sense. There are observations that majority of people tend to choose more popular product, than less popular, despite both being of a similar quality. The article [2] also cites numerous works, which speculate that herding behavior might be related to the fluctuations of asset price. In Kirman's formalization the switching probabilities do not depend on the source, thus probability distribution of ant's visiting times at both mangers is symmetric.
Kirman’s model and similar approaches have been applied as models of herding and contagion phenomena in financial markets [1,4,5]. In ref. [6] the equilibrium distribution of the related discrete-time stochastic process within the more general theoretical framework of Polya urn processes has been derived. The associated Fokker-Planck equation for the pertinent continuous symmetric dynamic process has been derived and solved in ref. [7]. The parameters of Kirman’s herding model applied to the description of financial markets were estimated in ref. [8] by introduction of a simulated moment approach extracting two key parameters of the model via matching of the empirical kurtosis and the first autocorrelation coefficient of squared returns. A direct estimation of the parameters of a related agent-based model, based on a closed-form solution of the unconditional distribution of returns, has been proposed in ref. [9]. The Kirman model was generalized in ref. [10]. It is worth to notice that the appropriate agent-based models can yield emergence the power-law scaling, long-range correlations, (multi)fractality and fat tails (see, e.g., [11,12] and references herein), however the omnipresent 1/f noise have not yet been revealed in such approach.
The phrases “1/f noise”, “1/f fluctuations”, and “flicker noise” refer to the phenomenon, having the power spectral density at low frequencies f of signals of the form S(f ) ∼ 1/f β , with β being a system-dependent parameter. Power-law distributions of spectra of signals with 0.5 < β < 1.5, as well as scaling behavior in general, are ubiquitous in physics and in many other fields, including natural phenomena, human activities, traffics in computer networks and financial markets [13][14][15][16][17][18][19][20][21][22][23]. This subject has been a hot research topic for many decades (see, e.g., a bibliographic list of papers by Li [24], and a short review in Scholarpedia [13]). Despite the numerous models and theories proposed since its discovery more than 80 years ago [25,26], the subject of 1/f noise remains still open for new discoveries. Most of the models and theories are not universal because of the assumptions specific to the problem under consideration. In 1987 Bak et al [27] introduced the notion of self-organized criticality (SOC) with the motivation to explain the universality of 1/f noise, as well. Although the paper [27] is the most cited paper in the field of 1/f noise problems, it was shown later on [28,29] that the mechanism proposed in [27] results in 1/f β fluctuations with 1.5 < β < 2 and does not explain the omnipresence of 1/f noise. On the other hand, we can point to a recent paper [30] where an example of 1/f noise in the classical sandpile model has been provided. A short categorization of the theories and models of 1/f noise is presented in the introduction of the paper [31].
Recently, the nonlinear stochastic differential equations (SDEs) generating signals with 1/f noise were obtained in Refs. [32,33] (see also recent paper [31]), starting from the point process model of 1/f noise [34][35][36][37][38][39]. Nonlinear SDEs provide macroscopic description of a complex system. Microscopic, agent based reasoning of equations exhibiting 1/f noise can yield further insights into behavior of the system. In this paper we show that it is possible to obtain nonlinear SDE of the form of Refs. [32,33] starting from agent-based herding model. Thus, it is possible to show analytically that nonlinear nature of herding interactions and appropriate choice of variable results in 1/f fluctuations.
The herding model. -In papers [1,2,4] Kirman employed a simple model to describe the behavior of a multitude of heterogeneous interacting agents. In the model the dynamic evolution is described as a Markov chain. There is a f
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