Information Technology Reference
In-Depth Information
WS decided to evaluate the statistical properties of these three different kinds of real
webs. They evaluated the degree of clustering in each web according to their definition
of a clustering coefficient
C
(
6.81
). They also evaluated the property
D
=
L
ij
,
(6.91)
where
L
ij
measures the minimal separation distance between two vertices in the graph.
D
is the average over all possible pairs.
WS compared the results of their observations on the three real networks with the
theoretical predictions of the random-graph theory for webs with the same numbers of
nodes. The mean number of links per node is given by
2
L
random
k
=
N
,
(6.92)
where
L
random
is the number of links that are realized in a random web with
probability
p
.Wehave
L
random
L
max
,
p
=
(6.93)
where
N
(
N
−
1
)
L
max
=
.
(6.94)
2
As pointed out earlier, the factor of 2 in (
6.92
) takes into account that each edge con-
tributes to the number of edges of the node
i
and to the number of edges of the node
j
at the same time. From (
6.93
)
L
random
=
pL
max
,
(6.95)
so that inserting (
6.95
)into(
6.92
) yields the average number of connections
2
pL
max
N
.
k
=
(6.96)
Finally, by inserting (
6.94
)into(
6.96
), we obtain
k
=
p
(
N
−
1
),
(6.97)
in accordance with what was found earlier.
In summary, then, using (
6.97
) we determine the values of the probability
p
of the
three webs under the assumption that they are random. We have for the film actors
61
222
p
=
525
=
0
.
000274127
,
(6.98)
,
for the power-grid networks
2
.
67
p
=
940
=
0
.
00054
(6.99)
4
,
and for the
C. elegans
network
14
281
=
p
=
0
.
0498
.
(6.100)
