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experimentally. The experimental observation of biological networks reveals [
25
] that
α
1 and 3. For this reason, AB [
3
] proposed a modified version
of the model of preferential attachment. They start with
m
0
isolated nodes, and at each
step they perform one of the following three operations.
is scattered between 2
.
m
0
) new links. For each link one node
is selected randomly and the other in accord with the criterion of preferential
attachment. This process is repeated
m
times.
(ii) With probability
q
AB rewire
m
links. They select randomly a node
i
and a link
l
ij
connected to it. Then they replace it with a new link
l
ij
that connects node
i
with
node
j
selected in accord with the preferential attachment criterion. This process
is repeated
m
times.
(iii) With probability 1
(i) With probability
p
they add
m
(
m
≤
q
AB add a new node. The new node has
m
new links
that are connected to the nodes already present in the web that had been selected
in accord with the preferential attachment criterion.
−
p
−
that can be either an inverse power
law or an exponential function. In Figure
6.20
examples of inverse power-law and expo-
nential distributions are depicted for the various conditions. It is important to point out
that AB prove that, in the scale-free regime,
Using this model AB realize a distribution
θ(
k
)
changes from 2 to 3. Figure
6.20
depicts
the comparison between the numerical simulations and the predictions made using con-
tinuum theory. The dashed lines in Figure
6.20
(a) illustrate the theoretical prediction
in the scale-free region. In Figure
6.20
(b) the exponential region is depicted; here the
distribution is graphed on semilogarithmic paper, converging to an exponential in the
q
α
→
1 limit.
(a) In the simulations
t
=
10
,
000 and
m
0
=
m
=
2. Circles,
p
=
0
.
3
,
q
=
0; squares,
p
=
0
.
6
,
q
=
0
.
1; and diamonds,
p
=
0
.
8
,
q
=
0. The data were logarithmically binned.
(b) Circles,
p
=
0
,
q
=
0
.
8; squares,
p
=
0
,
q
=
0
.
95; and diamonds,
p
=
0
,
q
=
0
.
99.
From [
29
] with permission.
Figure 6.20.
