Information Technology Reference
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to
k
takes into account all possible numbers of links. By inserting the Poisson
distribution (
6.63
)into(
6.69
) we obtain the transcendental equation for the probability
=∞
e
z
(
s
−
1
)
.
s
=
(6.70)
Using the definition (
6.68
) we transform (
6.70
) into the transcendental equation for the
fraction of the graph occupied by the giant cluster,
e
−
zS
S
=
1
−
.
(6.71)
In this equation
z
is a control parameter, or, using (
6.53
), it is proportional to the
probability
p
. To establish the critical value of the control parameter in this case, we
assume that
p
c
is the critical value of the probability compatible with the emergence
of a very small, but finite, value of
S
.For
S
very small we expand the exponential and
(
6.71
) becomes
S
=
zS
,
(6.72)
which indicates that the critical value is
z
c
=
1, thereby yielding, on the basis of (
6.53
),
1
p
c
=
1
.
(6.73)
N
−
This result is interesting, because it indicates that for very large numbers of nodes
N
the critical value of
p
or
z
may be extremely small. In Figure
6.9
we sketch the shape
of this phase-transition process.
The length of the random web
Let us now define another important property of random webs. This is the length
l
of
the web. How many nodes do we have to encounter to move from a generic node
i
to
another generic node
j
? An intuitive answer to this question can be given by assuming
that each node has a very large number of links that we identify with the mean value
z
1. This is equivalent to making the assumption that we are far beyond the threshold
value
z
c
=
1.
Each node of a network has
z
neighbors (
z
5 in the case of Figure
6.10
). Thus, in
the first layer there are
z
points connected to
P
. In the second layer there are
z
2
points.
In the
l
th layer we have for the number of points ultimately connected to
P
=
z
l
N
l
=
.
(6.74)
Figure 6.9.
A sketch of the phase-transition process created by increasing the control parameter
z
.
