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

This work involves an investigation of the mechanics of the herding behaviour using a non-linear timescale, with the aim to generalize the herding model which helps to explain frequently occurring complex behaviour in the real world, such as the financial markets. A herding model with fractional order of derivatives was developed. This model involves the use of derivatives of order α where 0<α ≤1. We have found the generalized result that the number of groups of agents with size k increases linearly with time as nk={p(2p-1)(2-α)/[p(1-α)+1]}Γ(α+(2-α)/(1-p){Γ(k)/[Γ(k-1+α+(2-α)/(1-p))}t for α ∈ (0,1], where p is a growth parameter. The result reduces to that in a previous herding model with 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 differentiation and fractional integration is also explored based on the results of this work.