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has to make a decision on its own, since in this case the coupling factor is one and does
not exert any influence on the site.
The key point is that the master equation for each site is perfectly well defined, but it
is a fluctuating master equation. In fact, the transition coefficients
g
ij
(
t
)
depend on the
π
r
, which have random values depending on the stochastic time evolution of
the environment at a specific site. As a matter of fact, we may define another ratio
quantities
J
r
J
,
r
=
(7.14)
=
where
J
denotes the total number of sites, namely
J
7 in the example of Figure
7.1
.
It is evident that the quantity
r
is also an erratic function of time, even if it is expected
to be smoother than
π
r
. In fact,
r
is a global property, obtained from the observation
of the whole network, whereas
π
r
is a property of the environment at a given site. The
smaller the cluster, the more erratic the quantity
π
r
.
One of the web properties of interest is the cooperation among the nodes and the con-
ditions under which we can expect
perfect consensus
. In the case of perfect consensus
all the nodes are in either the state
r
2 and do not jump from one
state to the other. The condition for
imperfect consensus
, by contrast, corresponds to the
collection of nodes making random jumps from one state to the other. The condition for
a lack of consensus corresponds to the coupling parameter being below some specified
value
K
=
1 or the state
r
=
K
c
, where
K
c
is the critical value of the control parameter below which the
flipping of the nodes between states is uncoordinated. In this case there are uncorrelated
fluctuations around the vanishing mean value. For
K
<
K
c
, on the other hand, there are
fluctuations around two distinct non-vanishing values and the web after a long sojourn
in one of the two states makes an abrupt transition to the other state.
Imagine that a web topology necessary to produce perfect consensus exists, in which
case the probability of being in a given state is
>
p
r
(
t
)
=
π
r
(
t
)
=
r
(
t
)
;
r
=
1
,
2
.
(7.15)
With this assumption we also set the condition of there being no difference from site
to site in the network. In a real case, however,
p
r
(
does change from site to site.
Consequently, when the condition of perfect consensus is realized all the sites obey the
same master equation (
7.9
) with the transition rates given by
t
)
g
ij
(
t
)
=
g
exp
[−
K
(
p
i
(
t
)
−
p
j
(
t
))
]
.
(7.16)
On introducing the difference variable
(
t
)
=
p
1
(
t
)
−
p
2
(
t
)
(7.17)
the master equation simplifies to
d
dt
=−
(
g
12
+
g
21
)
−
(
g
12
−
g
21
).
(7.18)
