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Figure 6.21.
Construction of the deterministic scale-free network, showing the first four steps of the iterative
process. Adapted from [
7
].
2
3
3
n
−
i
N
R
=
.
(6.108)
For instance, at
n
2 is six, in accordance
with (
6.108
). Let us imagine that we iterate the procedure up to time
n
=
3 the number of nodes with degree
k
R
=
1. Let us
focus on
i
1. Thus (
6.111
) yields
2
i
k
=
,
(6.109)
from which we get
ln
k
ln 2
.
i
=
(6.110)
It is easy to assess that the root, namely the hub with the highest degree, after
i
iterations
attains the degree
2
i
+
1
k
R
=
−
2
.
(6.111)
Taking the logarithm of (
6.108
) yields
ln
2
ln
N
R
=
3
+
(
n
−
i
)
ln 3
.
(6.112)
On inserting (
6.110
)into(
6.112
)wehave
ln
2
3
3
n
ln
1
k
α
ln
2
ln 3
ln 2
ln
k
ln
N
R
=
3
+
n
ln 3
−
=
+
,
(6.113)
with the power-law index
ln 3
ln 2
.
α
=
(6.114)
