Information Technology Reference
In-Depth Information
Thus, for a web with infinitely many nodes, the temporal behavior of the global web
variable
.
When the mean-field limit is not taken, there is a finite number of nodes and the
dynamical picture stemming from the above master equation is changed. Bianco et al .
[ 15 ] pointed out that in the finite-number case the master equation is that for the infinite-
number situation except that the transition rates fluctuate,
ξ(
t
)
is identical to that of the dynamical variable
(
t
)
(7.23)
where the fluctuations are on the order of 1/ N as we found for the random-walk
process. If the number of nodes is very large, but still finite, we consider the mean-
field approximation to be nearly valid and are able to replace the deterministic equation
( 7.19 ) with the stochastic equation
d
g ij
g ij + ε ij ,
(
t
)
=−
U
()
η(
t
)(
t
) + ε(
t
),
(7.24)
dt
where the multiplicative fluctuation has the form
η(
t
) = ε 12 (
t
) + ε 21 (
t
)
(7.25)
and the additive fluctuation is given by
ε(
t
) = ε 12 (
t
) ε 21 (
t
).
(7.26)
The random fluctuations induce transitions between the two states of the potential well.
Thus, for a web with a finite number of nodes the phase synchronization of ( 7.19 ) is not
stable.
A simpler stochastic equation is obtained if the fluctuations are anti-symmetric;
that is, if
ε 12 (
) =− ε 21 (
)
t
t
. In this simpler case the multiplicative coefficient van-
ishes,
η(
t
) =
0
,
and ( 7.24 ) reduces to the Langevin equation with additive random
fluctuations,
d
ξ(
t
)
=−
U
(ξ)
∂ξ
+ ε(
t
),
(7.27)
dt
using ( 7.22 ) for the web variable. Equation ( 7.27 ) represents the Smoluchosky approx-
imation to the physical double-well potential discussed in Chapter 3 with the addition
of a random force. The fluctuations drive the particle from one well of the potential
to the other by giving it sufficient amplitude to traverse the barrier between the wells.
However, here the fluctuations arise from the finite number of nodes in the web rather
than from the thermal behavior of a heat bath.
The global variable fluctuates between the two minima as described by ( 7.27 )for
values of the coupling parameter greater than the critical value as depicted in Figure 7.4 .
The single node follows the fluctuations of the global variable, switching back and forth
from the condition where the state r
=
1 is preferred statistically to that where the state
r
2 is preferred statistically. In physics this is similar to what occurs in the Ising model
of magnetization, where each element has either up or down spin and spins are allowed
to interact with one another. For constant-strength interactions the material undergoes
a phase transition, going from a state of no magnetization with each spin fluctuating
=
Search WWH ::




Custom Search