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the lhs of which is the Laplace transform of the time derivative of the probability and
the rhs of which is the convolution of the product of the memory kernel and the prob-
ability density. Consequently, taking the inverse Laplace transform of ( 7.50 ) yields the
generalized master equation (GME)
d p
t
(
t
)
dt (
t )
t ).
=
t
Kp
(
(7.51)
dt
0
Note that the GME is non-Markov except when the memory kernel is a delta function in
time. The derivation of the GME rests on the hypothesis that the dynamics of complex
webs are determined by the action of events, and that it is necessary to assume that these
events are crucial to account for the deviation from ordinary statistical physics revealed
by real experiments; see for example [ 5 , 57 ].
7.2.2
Relaxation of the two-level GME
Many continuous phenomena are well modeled by two states; for example we have the
on and off states of blinking quantum dots, a form of physical network; in biology there
is the respiratory network with the inhalation and exhalation of the lungs; in informa-
tion theory there is the connectivity of the Internet; in sociology there is the two-state
decision-making process, say voting for a candidate; we can use two-state models to
capture the dynamics of any web characterized by deciding between two alternatives.
In the situation where N
2 we can reduce the GME given by ( 7.51 ) to two coupled
integro-differential equations,
dp 1 (
=
t
t
)
1
2
dt (
t ) [
t )
t ) ] ,
=−
t
p 1 (
p 2 (
dt
0
t
dp 2 (
t
)
1
2
dt (
t ) [
t )
t ) ] ,
=+
t
p 1 (
p 2 (
(7.52)
dt
0
where we have used the form of the transition matrix given by ( 7.8 ). On taking the
difference between these two equations and again introducing the difference variable
the two-state GME reduces to the compact form
t
d
(
t
)
dt (
t )(
t ).
=−
t
(7.53)
dt
0
An interesting general relation results from the solution to the equation for the
difference variable. The Laplace transform of ( 7.53 ) yields
u (
) =− (
)(
) (
),
u
0
u
u
which, after some rearrangement, gives
(
)
0
(
) .
However, replacing the memory kernel in this equation using ( 7.49 ) results in
) =
u
+ (
u
u
ψ(
1
u
)
(
) = (
u
) =
(
0
u
)(
0
),
(7.54)
u
 
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