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scaling of webs within organisms. Scaling and allometric growth laws are inextricably
intertwined, even though the latter were introduced into the biology of the nineteenth
century and the former gained ascendancy in engineering in the latter part of the twen-
tieth century. There is no uniform agreement regarding the relation between the two,
however. Calder pointed out that it would be “a bit absurd” to require the coefficient
α
in (
1.44
) to incorporate all the dimensions necessary for dimensional consistency of the
allometric equation. He asserts that
The allometric equations
...
are empirical descriptions and no more, and therefore qualify for
the exemption from dimensional consistency in that they relate some quantitative aspect of an
animal's physiology, form, or natural history to its body mass, usually in the absence of theory or
prior knowledge of causation.
Allometric growth laws are analytic expressions relating the growth of an organ to the
growth of an organism of which the organ is a part, but historically they do not have a
dynamical foundation. A contemporary of Lotka was Julian Huxley, who did attempt to
put allometric relations on a dynamical basis by interrelating two distinct equations of
growth. In the simplest form we write two growth laws for parts of the same organism as
dW
b
dt
=
bW
b
,
dW
a
dt
=
aW
a
,
and, on taking the ratios of the rate equations, obtain
dW
b
W
b
=
b
a
dW
a
W
a
.
(1.47)
This is just the equation obtained from the allometric relation (
1.45
), from which we
see that the parameters can be related by
a
b
β
=
(1.48)
so that the allometric index is given by the ratio of the growth rate of the entire body to
that of the member organ. In his topic
Problems of Relative Growth
[
16
] Huxley records
data ranging from the claws of fiddler crabs to the heat of combustion relative to body
weight in larval mealworms, all of which satisfy allometric growth laws.
Some forty years after Lotka, the physicist de Solla Price [
9
]pickeduponthefor-
mer's study of scientific publications, extended it, and founded a new area of social
investigation through examination of the number of citations of scientific papers during
varying intervals of time. Figure
1.11
depicts the number of papers published having
a given number of citations. Let us stop a moment and interpret this distribution. It is
evident that the first 35% of all scientific papers published have no citations, the next
49% have one citation, there are two citations for the next 9% and so on. By the time the
average number of 3.2 citations per year is reached, 96% of the distribution has been
exhausted. It should be clear from this that the average number of citations does not
characterize this social web.
