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discrete location on a lattice at a given time; a multivariate one-time probability is the
probability of multiple events occurring at various sites but at a specific time; a con-
ditional probability depends on two lattice sites and two times, the occurrence of an
event at the latter location and time being conditional on the occurrence of an event at
the former location and earlier time. All these various technical definitions of probabil-
ities were given full voice in Markov's theory at the turn of the last century, but, other
than acknowledging the existence of this body of theory, we find that much of what
we need to understand about the statistics of complex webs lies outside the domain of
Markov theory. To clearly label this difference the complex webs discussed fall under
the heading of non-Markov phenomena.
7.1.1
Discrete dynamics
The master equation was first written under that name as a differential-difference equa-
tion to describe gain-loss processes [
50
]. In its simplest representation the master
equation can be written in matrix form, where
p
=
(
p
1
,
p
2
,...,
p
N
)
is a probability
vector and
K
is an
N
×
N
matrix of time-independent transfer coefficients,
d
p
(
t
)
=
Kp
(
t
),
(7.1)
dt
which clearly has the formal solution
e
K
t
p
p
(
t
)
=
(
0
).
(7.2)
The behavior of the probability density is therefore completely determined by the eigen-
value spectrum of the transition matrix. Suppose that
S
is the matrix that diagonalizes
the transition matrix, yielding
⎛
⎝
⎞
⎠
λ
0
000
•
0
λ
1
0
SKS
−
1
=
=
00
•
0
,
(7.3)
0
•
•
λ
N
−
1
where the eigenvalues are ordered
|
λ
0
|
<
|
λ
1
|
< ... <
|
λ
N
−
1
|
for a finite lattice of size
N
. The general solution (
7.2
) can then be written as
N
−
1
C
j
e
λ
j
t
p
(
t
)
=
C
0
+
,
(7.4)
j
=
1
where the
C
-vectors are expressed in terms of the similarity-transform elements and
the initial condition on the probability vector. In order for the asymptotic probability
density to be a constant vector, that is, a steady-state solution to the master equation
lim
p
(
t
)
=
C
0
,
(7.5)
t
→∞
the lowest eigenvalue must vanish,
0, and the real parts of the remaining eigenval-
ues must be negative. These are the constraints on the eigenvalues and consequently on
λ
0
=
