Webs - Complex Webs: Anticipating the Improbable - page 24

log y + β log x − b = 0 (1.42)

or, in more compact form,

yx β = constant. (1.43)

Equation ( 1.43 ) is a generalized hyperbolic relation between genera and number of
species and covers a wide variety of cases including both plants and animals.
As noted by Lotka, the monotypic genera, with one species each, are always the most
numerous; the ditypics, with two species each, are next in rank; genera with higher
numbers of species become increasingly fewer with increasing number. In Figure 1.7 it
is clear that the fitting curve is an inverse power law with slope . The pure inverse

power law was found to be true for all flowering plants and for certain families of
beetles as well as for many other groupings of plants and animals. Note that the pure
inverse power law is the asymptotic form of the distribution resulting from entropy
maximization when the logarithm is used as the constraint to impose scaling on the
random variable. Moreover, what is meant by asymptotic is determined by the size of
the constant T in ( 1.41 ).
Lotka did not restrict his observations to biodiversity; his complexity interests
extended to the spreading of humans over the Earth's surface in the form of urban
concentration. The build up of urban concentration had been investigated earlier by
F. Auerbach [ 2 ], who recognized that, on ordering the cities of a given country in order
of decreasing size from the largest to the smallest, the product of the city's size and the
city's rank order in the sequence is approximately constant, as given by ( 1.43 ). Lotka
applied Auerbach's reasoning to the cities in the United States, as shown in Figure 1.8 ,
which is a log-log graph of population versus rank order of the 100 largest cities in the
United States in a succession of years from 1790 to 1930. It is apparent from this figure
that, for the straight-line segment given by ( 1.42 ), where y is the city population and x is
the city rank, the empirical parameter b changes from year to year, but the power-law
index

− β

Lotka leaves open the question of the signifi-
cance of this empirical relation, although he does give some suggestive discussion of the
properties of certain thermodynamic relations that behave in similar ways. In the end
Lotka maintains that he does not know how to interpret these empirical relations, but
subsequently he constructed an empirical relation of his own in a very different context.
As an academic Lotka was also curious about various measures of scientific produc-
tivity and whether or not they revealed any underlying patterns in the social activity
of scientists. Whatever his reasons, he was the first to examine the relation between
the number of papers published and the number of scientists publishing that number of
papers. This relationship provides a distribution of scientific productivity that is not very
different from biodiversity and the urban-concentration context just discussed. Lotka's
law,showninFigure 1.9 , is also an empirical relation that connects the fraction of
the total number of scientists publishing a given number of papers with the number of
papers published in order of least frequent to most frequent. Here again, as in the earlier
examples, the distribution is an inverse power law in the number of papers published
but with a power-law index

β = 0 . 92 remains fairly constant .

β ≈ 2. From Figure 1.9 we see that for every 100 authors