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ln N
=
) .
L
(6.128)
ln
(
ln N
Using ( 6.78 ) we may conclude that the mean number of links is given by z
ln N .The
same value for the length of the preferential attachment network has more recently been
derived by Chen et al. [ 12 ], who confirmed that ( 6.128 ) holds for any value of m (the
rigorous demonstration of ( 6.128 ) was given by [ 9 ] only for m
=
.
In 2003 Cohen and Havlin [ 13 ] studied the problem of the relation between
=
1
)
α
and L ,
and found that for
3, which is the boundary between the region where the mean-
square number of connections is finite (
α =
α>
3),
k 2
< ,
and the region where the mean-square number of connections diverges (
α<
3),
k 2
=∞ ,
the web has the following length:
ln N
L
) .
(6.129)
ln
(
ln N
They also found that for
2
<α<
3
(6.130)
the web length is given by
L
ln
(
ln N
).
(6.131)
For this reason these webs are considered to be ultra-small.
It is evident that the smallness of a web is important in order for it to carry out its
function. This is especially important for the Internet. If the length of the web increased
linearly with the addition of new sites, the excess information would make the Inter-
net useless. The fact that the Internet is scale-free yields L
ln N , thereby strongly
reducing the size increase with increasing number of nodes N . In the special case
α<
3,
the increase of L with N is further reduced.
α
= 1
L = 2
Ultra-small networks
α
= 2
L = ln(ln N )
Small networks
α
= 3
L = ln N /ln(ln N )
L = ln N
Ordinary Graphs
α
> 3
This figure shows the relation between the scale-free property of the web and the web size L .
Figure 6.24.
 
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