Abstract
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
n
k
=
p
(
2
p
−
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 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.