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the elements of the transition matrix that make the master equation an acceptable model
of the phenomenon being investigated. Of course there are mathematical subtleties that
confound this straightforward interpretation, such as the situation when more than one
eigenvalue is zero, in which case the exponential relaxation of the web to an asymptotic
steady state might not occur. Other, more complex, forms of relaxation may arise; see
Oppenheim
et al.
[
51
] for a more rigorous mathematical discussion of solutions to the
master equation and a presentation of the background literature.
For illustrative purposes, it is convenient to consider the case in which the unit time
is
t
=
1. Thus, we write a discrete version of (
7.1
) in the form of an iterative equation
p
(
n
+
1
)
=
(
I
+
K
)
p
(
n
),
(7.6)
where
I
is the
N
N
unit matrix. Let us assume that at any time step
n
we toss a
coin to select the state occupied by the network at time
n
×
1. It is evident that such a
process reduces the dimensionality of the master equation to two, changing the generic
vector to
p
+
2
,
2
since
1
(
n
)
=
(
p
1
(
n
),
p
2
(
n
))
into the equilibrium condition
p
(
n
)
=
p
1
+
p
2
=
1
.
Consequently, the coefficient of the preceding state can be written as
2
1
2
(
I
+
K
)
=
,
(7.7)
1
2
1
2
where
I
is the 2
×
2 unit matrix and the transition matrix becomes
1
2
1
2
−
K
=
.
(7.8)
1
2
1
2
−
This form of the transition matrix is typical of the classical master equation.
7.1.2
A synchronized web of two-level nodes
Consider the master equation to describe the dynamics of the two-level node obtained
from the equations with which we ended the last subsection:
1
dp
1
dt
=−
g
12
p
1
+
g
21
p
2
,
(7.9)
dp
2
dt
=−
g
21
p
1
+
g
12
p
2
.
×
Here the elements of the 2
2 coupling matrix
K
are the
g
s, which in the simplest
case of a coin flip are all 1/2. When the matrix elements are all equal
g
21
=
g
12
=
g
and this equation can be derived from the subordination to the coin-tossing prescription
discussed earlier, by using the subordination function
re
−
r
τ
,
ψ(τ)
=
(7.10)
so that
g
21
=
Now we generalize the model and embed this single two-
state node into a web of such nodes as indicated schematically in Figure
7.1
.
g
12
=
g
=
r
/
2
.
1
Much of the discussion of this subsection is taken from [
15
].
