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for which the distribution is independent of time and the initial state of the web.
Imposing a zero-flux boundary condition yields
q
g
)
=
)
∂
∂
−
G
(
q
)
P
ss
(
q
)
+
Dg
(
q
(
q
)
P
ss
(
q
0
,
so that
1
∂
∂
G
(
q
)
q
[
g
(
q
)
P
ss
(
q
)
]=
2
.
(3.138)
g
(
q
)
P
ss
(
q
)
Dg
(
q
)
Integrating (
3.138
) yields the exact steady-state probability density
exp
1
D
G
q
)
dq
Z
(
P
ss
(
q
)
=
,
(3.139)
q
)
2
g
(
q
)
g
(
where
Z
is the normalization constant. It will prove useful to have this solution available
to us for later analysis of complex webs.
An example of multiplicative fluctuations is given by an open network whose connec-
tions are determined by a stochastic rate. Assume that the growth of a network, that is,
the number of connections of a node to other nodes, is determined by a combination of
a feedback that keeps the growth from becoming unstable and a preferential attachment
having a random strength such that
dk
(
t
)
=−
λ
k
(
t
)
+
ξ(
t
)
k
(
t
).
(3.140)
dt
Here
k
is the number of connections of the node of interest to other nodes in the web
at time
t
. We assume that the fluctuating rate is zero-centered, delta-correlated in time
and has Gaussian statistics. With these assumptions regarding the fluctuations we can
construct a FPE of the form (
3.136
) with the functions
(
t
)
G
(
k
)
=−
λ
k
(3.141)
and
g
(
k
)
=
k
.
(3.142)
The steady-state solution to this FPE is given by (
3.139
) and substituting (
3.141
) and
(
3.142
) into this expression yields the analytic solution
A
k
μ
,
P
ss
(
k
)
=
(3.143)
with the power-law index, given in terms of the parameters of this example,
+
D
.
μ
=
1
(3.144)
This solution and others from nonlinear multiplicative Langevin equations are discussed
by Lindenberg and West [
30
] as well as the conditions for normalization for
k
0.
Note that the inverse power-law solution to the FPE, when suitably restricted, agrees
with the preferential-attachment model of Barabási and Albert (BA) [
5
], which we
discuss later, if we take the parameters to be in the ratio
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