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n=0
n=1
n=2
i=1
i=2
n=3
i=3
i=4
Figure 6.23.
Construction of the deterministic scale-free network, showing the first four steps of the iterative
process. Adapted from [
11
].
co-workers. We expect that the network becomes more efficient if the root interacts
directly also with the subordinate co-workers of his/her two vice presidents. The same
remarks apply to his/her vice presidents. They do not interact only with their two direct
co-workers but also with the co-workers of the co-workers. The straight-line segments
at the level
n
2 indicate that the root interacts also with the co-workers of his/her
direct co-workers. The straight-line segments emanating from
i
=
=
1to
i
=
4atthelevel
n
3 show that the two vice presidents organize their interaction with their cascade of
co-workers according to the rules of this university. They interact directly also with the
co-workers of their co-workers. Note that the straight-line segments at the level
n
=
3
indicate that the president has a direct connection with all the nodes of the bottom layer.
The authors of this model prove numerically that
=
α
≈
1
.
(6.125)
They also prove analytically, but we omit the proof here, that the length of this web is
2
2
N
+
3
2
N
+
1
−
(
2
N
+
4
)
L
=
)
.
(6.126)
(
2
N
+
1
−
1
)(
2
N
+
1
−
2
Thus,
2
2
2
N
+
2
2
N
+
1
−
(
N
+
2
)
lim
N
L
(
N
)
=
lim
=
2
.
(6.127)
2
2
N
+
2
→∞
N
→∞
As a consequence, it is possible to move from any node
i
to any other node
j
in two steps
or more. This is really a very small world and next we argue why this is the smallest
possible world.
6.3.2
From small to ultra-small networks
Bollobás and Riordan [
9
] used rigorous mathematical arguments to derive the results of
BA, the case of preferential attachment illustrated earlier, and reached the conclusion
that the length of the BA network is
