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We write (
6.113
)as
ln
A
(
n
)
ln
N
R
=
,
(6.115)
k
α
with the coefficient
3
n
−
1
A
(
n
)
≡
2
·
.
(6.116)
In conclusion the probability density for connectivity of the nodes is, using the
number of nodes with
k
R
links,
1
k
α
,
θ(
k
)
∝
(6.117)
with
α
defined by (
6.114
), the numerical value of which is
1
.
099
α
≈
6939
≈
1
.
59
.
(6.118)
0
.
The above argument is similar in spirit to that given for determining the length of
a fractal curve connecting two points separated by a finite Euclidean distance. Let us
construct the curve obtained by adding a bit of length to the line segment connecting
the two points at each step of an iteration process. Consider a line segment of unit length
and partition it into three pieces of equal size. Now we replace the center piece with a
section that is twice the original length of that segment, thereby making a sharp cap in
this interval as shown in Figure
6.22
. The length of the curve therefore increases from
unity at
n
1. Now carry this process a step further and partition each
of these four line segments into thirds and place a cap in the center third of each one of
these as shown in Figure
6.22
for
n
=
0to4
/
3at
n
=
=
2. The total length of the curve at
n
=
2 is 16/9,
η
where each of the sixteen line segments is of length 1/9. In general, if
is the length
(η)
of the ruler and
N
is the number of times the ruler is laid end to end to measure the
length of the curve, the length
L
(η)
is given by
L
(η)
=
N
(η)η.
(6.119)
This relation is, of course, true in general.
In the construction process the size of the ruler is determined by the iteration number
and in the example given the ruler size is
3
n
at generation
n
. Consequently the
η
=
1
/
length of the curve at generation
n
is
N
)
3
n
.
(
n
L
(
n
)
=
(6.120)
The number of times the ruler is laid down to measure a curve of fractal dimension
D
is given by
N
)
=
η
−
D
, so the length given by (
6.120
)is
(
n
1
3
n
1
−
D
L
(
n
)
=
.
(6.121)
Moreover, the curve becomes incrementally longer with each iteration and the lengths
of the curve at successive generations are related by
