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<α⩽1 . We have found the generalized result which indicates that number of groups of agents with size k increases linearly with time as nk=p(2p−1)(2−α)p(1−α)+1Γ(α+2−α1−p)Γ(k)Γ(k−1+α+2−α1−p)t for α∈(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.