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and
μ>
1 since
p
<
1 and the scale factor
N
>
1. Consequently, the general solution
to the scaling equation can be written as
∞
H
n
G
(ξ)
=
A
n
ξ
,
(2.16)
n
=−∞
where the scaling index is the complex quantity
i
2
n
ln
N
.
π
H
n
=−
μ
+
(2.17)
given by (
2.14
) is simplified in that
this function reduces to a constant, cf. Figure 1.6. Montroll and Shlesinger [
50
] deter-
mined that the Pareto index is given by
In an economic context the general form of
A
(ξ)
μ
≈
1
.
63 for the 1935-36 income distribution
in the United States. If
pN
represents the distribution
function of a divergent branching (bifurcating) process in which the mean value is infi-
nite. If
pN
>
1, then 1
<μ<
2 and
G
(ξ)
1but
pN
2
is finite.
However, the measure of the fluctuations about the mean value, namely the standard
deviation, is infinite. These various ranges of the index are determined from the data.
Using the empirical value determined above, if we choose the probability of being in
the amplifier class to be
p
<
>
1
,
then 2
<μ<
3 and the mean value of
G
(ξ)
01, then the amplification factor
N
would be approx-
imately 16.8. As pointed out by Montroll and Shlesinger [
50
], this number is not
surprising since one of the most common modes of significant income amplification
in the United States is to organize a modest-sized business with on the order of 15-20
employees. This might also suggest the partitioning between relatively simple and truly
complex networks, but that would lean too heavily on speculation.
It is clear that this argument is one possible explanation of the inverse power-law tail
observed in Figure
2.9
and, using the data from Dodds and Rothman [
18
], the average
value of the power-law index is estimated to be
=
0
.
This value of the scaling
index is not too different from that obtained in the economic application. This kind
of argument falls under the heading of renormalization and is taken up in some detail
later because it is a powerful tool for identifying scaling fluctuations in erratic time
series.
The geophysical applications of da Vinci's ideas are interesting, and in the next
section we develop them further in a physiologic context. Da Vinci was quite aware
that his notions were equally applicable in multiple domains, from the sap flowing up
the trunk of a tree out to the limbs, to the water flowing in a branching stream. However,
let us take a step back from the renormalization scaling argument used above and con-
sider what classical scaling yields. Before we do this, however, it is interesting to see
how his ideas reemerged in a variety of contexts from physiology to geophysics and are
among the empirical laws recorded in Tables
2.1
and
2.2
where the disciplines of geo-
physics, physiology, economics, sociology, botany and biology have all had their share
of inverse power laws. It should also be pointed out that in some phenomena recorded
in these tables a better form for the distribution of the web variable is the hyperbolic,
which is asymptotically an inverse power law.
μ
≈
1
.
83
.
