Information Technology Reference
In-Depth Information
Table 6.1.
Derived from [
35
], with the value of 0.005 for
C
random
of the power
grid probably being a misprint, which is here replaced with the value 0.0005
L
actual
L
random
C
actual
C
random
Film actors
3
.
65
2
.
99
0
.
79
0
.
00027
Power grid
18
.
7
12
.
4
0
.
08
0
.
0005
C. elegans
2
.
65
2
.
25
0
.
28
0
.
05
Insofar as the distance
L
is concerned, according to (
6.78
) we can write
ln
N
ln
L
random
=
)
,
(6.101)
(
k
which yields for the film-actor web
ln
(
222
,
526
)
12
.
31
L
random
=
=
11
=
2
.
99
,
(6.102)
ln
(
61
)
4
.
for the power-grid web
ln
(
4
,
941
)
8
.
51
L
random
=
)
=
98
=
8
.
68
(6.103)
ln
(
2
.
67
0
.
and for the
C. elegans
web
ln
(
282
)
5
.
64
L
random
=
)
=
63
=
2
.
14
.
(6.104)
ln
(
14
2
.
Note that there seems to be an ordering from the most human-designed web with the
highest clustering to the least human-designed, or, said differently, the most naturally
evolved web has the lowest clustering. Insofar as
C
is concerned, using (
6.87
) we expect
C
random
=
p
.
(6.105)
From Table
6.1
we get some important information. Both random and real webs have
short distances. In other words, the small-world property is shared by both actual and
random webs. The most significant difference between theory and data is given by the
clustering coefficient
C
. The real webs have large clustering coefficients, whereas the
random webs have small clustering coefficients.
WS [
35
] claim that real webs are complex in the sense that they exist in a condi-
tion intermediate between randomness and order. They also propose a model to create
networks with the same statistical properties as real networks. With probability
p
they
rewire some of the links established according to the deterministic prescription. Note
that the probability
p
adopted by them to rewire their links should not be confused with
the probability
p
used in Section
6.2.1
. In fact, when WS rewire a link, they adopt the
criterion of uniform probability that is equivalent to setting
p
of Section
6.2.1
equal to
the inverse of the number of possible new links.
Note that when
p
0 and the
regular
condition of Figure
6.17
applies, all the
triples are triangles. Thus,
C
=
=
1. When
p
=
1, all the links are established randomly.
