Growth and addition in a herding model with fractional orders of derivatives

Abstract

This work involves an investigation of the mechanics of the herding behavior using a non-linear timescale, with the aim to generalize the herding model which helps to explain frequently occurring complex behavior in the real world, such as the financial markets. A herding model with fractional orders of derivatives was developed. This model involves the use of derivatives of order α where $0 \lt \alpha\unicode{x2A7D}1$ . We have found the generalized result which indicates that number of groups of agents with size k increases linearly with time as $n_{k} = \frac{p(2p-1)(2-\alpha)}{p(1-\alpha)+1}\Gamma\left(\alpha+\frac{2-\alpha}{1-p}\right)\frac{\Gamma(k)}{\Gamma\left(k-1+\alpha+\frac{2-\alpha}{1-p}\right)}t$ for $\alpha \in (0,1]$ , where p is a growth parameter. The result reduces to that in a previous herding model with a derivative order of 1 for α = 1. The results corresponding to various values of α and p are presented. The group-size distribution at long time is found to decay as a generalized power law, with an exponent depending on both α and p , thereby demonstrating that the scale invariance property of a complex system holds regardless of the order of the derivatives. The physical interpretation of fractional calculus is also explored based on the results of this work.

Keywords

• growth and addition
• herding model
• fractional calculus
• fractional derivative
• fractional integral
• generalized power law