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8,000
1,000
100
10
2.5
Rank
10
50
100
1,000
Figure 1.8.
Auerbach's law. The populations of United States cities are graphed as the ordinate and on the
abscissa is the corresponding city rank in order of magnitude, both on logarithmic scales [
42
].
who publish a single paper in a given period of time there are 25 authors who publish
two papers, 11 who publish three, and so on. Do not be concerned about the details of
this interpretation for the moment; how the interpretation is determined will be laid bare
subsequently.
These last two examples involve social webs. The former has to do with how people
interact with one another to form communities and how these communities aggregate to
form towns, which in turn merge to form cities. It seems that most individuals are more
strongly attracted to regions where others have already settled, but it is the nature of the
inverse power law that the hermit who wants no part of society is also accommodated by
the distribution. It might even be possible to construct a theory in which the attraction to
these regions depends on the local concentration density of people, but developing such
a theory would be getting ahead of ourselves. It is worth mentioning now, however, that
sociologists refer to this in a modern context as
Matthew's effect
, for reasons that will
eventually be clear. We return to this question in later chapters. The second social web
discussed has to do with the requirements to publish a paper and these requirements
will be discussed subsequently. However, these are not the only, or even the dominant,
phenomena that are described by inverse power laws. Such distributions also occur in
many different kinds of biological phenomena.
The complex interrelation in biological development between form and function was
addressed at the turn of the last century in the seminal work of D'Arcy Thompson [
33
].
He argued that the elements of biological webs are physical and therefore subject to
physical laws, but the fact that they are animate implies rules that may be distinct from
those rules that govern inanimate webs. The growth of biological webs is therefore
