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no. of species
10
1
30
100
2
0
0
0.2
0.4
0.6
0.8 1.0
log (no. of species)
1.2
1.4
1.6
1.8
2
Figure 1.7.
Willis' law. The relation between the number and size of Rubiaceae compared with the slope of
the fit to all flowering plants [
18
].
average, but the majority making a fair living. Pareto observed that in stable societies
that is not what occurs; there is a fundamental imbalance in how income is distributed.
He came to believe that the inverse power-law distribution of income, which subse-
quently became known as Pareto's law, was the expression of a social law. This was
the first glimpse into how complex webs organize their complexity or, alternatively, are
organized by their complexity.
Over the last century a substantial number of topics addressing the complexity of
webs has been written, and we shall draw from as many of them as we need in our
discussions. Lotka's topic [
18
] on biological webs focused on the macroscopic level
using statistical physics of near-equilibrium phenomena as a modeling strategy, but he
was very sensitive to the data available to test various biophysical hypotheses. One data
set had been compiled by J. C. Willis, who had published a topic on evolutionary biology
[
40
] in which he (Willis) argued that the spatial area occupied by a biological species
is directly proportional to the age of the species, so that the area occupied becomes a
measure of evolutionary age. The details of Willis' 1922 theory do not concern us here,
in part because the critique of it contained in Lotka's topic was not particularly positive,
but Willis' data are still interesting. Willis collected data on a variety of natural families
of plants and animals and graphed the number of genera as the ordinate (
x
) and the
number of species in each genus (
y
) as the abscissa. Figure
1.7
depicts the number and
size of Rubiaceae
1
on log-log graph paper. The data are fit by a line segment satisfying
the linear equation in the logarithms of the variables
1
The Rubiaceae is a family of flowering plants consisting of trees, shrubs, and occasionally herbs comprising
about 650 genera and 13,000 species.
