Information Technology Reference
In-Depth Information
The probability that
k
k
j
can be expressed in terms of the probability density
introduced earlier in a different context,
>
k
0
θ(
k
)
dk
,
P
(
k
j
<
k
)
=
(6.14)
so that
dP
t
j
>
1
/β
t
m
k
j
dP
(
k
j
<
k
)
θ(
k
)
=
=
dk
dk
m
1
/β
k
1
+
1
/β
.
t
m
0
+
1
β
=
(6.15)
t
In the asymptotic limit and recalling that
β
=
1
/
2 we obtain for the probability density
2
m
2
k
3
,
θ(
k
)
=
(6.16)
which establishes
3 for what is called a scale-free network in the literature.
The scale-free nature of complex webs affords a single conceptual picture spanning
scales from those in the World Wide Web to those within an organization. As more peo-
ple are added to an organization, the number of connections between existing members
depends on how many links already exist. In this way the oldest members, those who
have had the most time to establish links, grow preferentially in connectedness. Thus,
some members of the organization have substantially more connections than do the
average, many more than predicted by any bell-shaped curve. These are the individuals
out in the tail of the distribution, the gregarious individuals who seem to know everyone.
Of course,
α
=
3 is not characteristic of most empirical data, so we subsequently return
to the model in a more general context as first discussed by Yule [
38
].
α
=
6.1.2
Communication webs
There has been a steady increase in the understanding of a variety of complex webs over
the past decade. The present resurgence of interest in the field began with the pioneer-
ing paper of Watts and Strogatz [
35
], who established that real-world networks deviate
significantly from the totally random webs considered half a century earlier [
15
]. They
observed that real networks exhibit significant clustering that was not observed in the
early theory and that these strongly interacting clusters were weakly coupled together,
leading to the small-world theory. Barabási and Albert (BA) pointed out that small-
world theory lacked web growth and introduced preferential attachment, resulting in a
complex network having an inverse power-law connectivity distribution density with
power-law index
3[
6
]. BA referred to such complex webs as scale-free because
the distribution in the number of connections
k
has the form
α
=
p
(λ
k
)
=
g
(λ)
p
(
k
)
which we discussed in Chapter 2. This relation indicates that a change in scale
λ
k
→
k
results in merely an overall change in amplitude
A
of the distribution
A
→
g
(λ)
A
.