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dynamics have a long time scale, as we saw in the discussion of booster dynamics. On
the other hand, for very fast dynamics, the memory kernel undergoes a rapid regression
to equilibrium and we recover the typical memoryless structure of the classical mas-
ter equation. To see this more clearly, let us suppose that the waiting-time distribution
density is exponential,
ge
−
gt
ψ(
t
)
=
,
(7.57)
with the Laplace transform
g
ψ(
u
)
=
u
.
(7.58)
g
+
On inserting the Laplace transform for the waiting-time density (
7.57
) into the Laplace
transform for the memory kernel (
7.49
) we obtain
(
)
=
,
u
g
which implies, using the inverse Laplace transform, that the memory kernel is a delta
function in time,
(
t
)
=
g
δ(
t
).
(7.59)
The delta-function memory kernel restricts the GME (
7.51
)tothesimplerform
d
p
(
t
)
=
g
Kp
(
t
).
(7.60)
dt
This master equation has the same Markov structure as does the original, only the
transition elements are scaled by the constant factor
g
.
So now we know that the subordination function is not exponential and we address
the question of how to describe the effect of a perturbation on the complex network.
It is reasonable to assume that the perturbation affects the parameters of the transition
matrix since these elements model the interactions between the states of the network.
Sokolov [
59
] has recently proposed that in the general case the GME should read
t
d
p
(
t
)
dt
(
t
)
t
)
=
t
−
K
(
t
)
p
(
(7.61)
dt
0
and this somewhat arbitrary generalization of the GME leads to some interesting results.
Here the perturbation is assumed to occur after the last event, that is, at time
t
and no
earlier. In the two-state case we write
−
K
+
(
t
)
+
K
−
(
t
)
K
(
t
)
=
(7.62)
+
K
+
(
t
)
−
K
−
(
t
)
and weakly perturb the matrix elements
1
2
[
K
±
(
)
=
∓
ξ
p
(
)
]
.
t
1
t
(7.63)
