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to be scale-free. We document this association subsequently, but now let us turn from
geometric fractals to fractal distributions.
One way to picture a distribution having a fractal scaling property is to imagine
approaching a mass distribution from a great distance. At first, the mass will seem to be
in a single great cluster. As one gets closer, the cluster is seen to be composed of smaller
clusters. However, upon approaching each smaller cluster, it is seen that each subcluster
is composed of a set of still smaller clusters and so on. It turns out that this appar-
ently contrived example in fact describes a number of physical, social and biological
phenomena that we discuss in due course.
Let us now look at the distribution of mass points depicted in Figure
2.15
. The total
mass
M
is proportional to
R
d
, where
d
is the dimension of the space occupied by
the masses. In the absence of facts to the contrary it is reasonable to assume that the
point masses are uniformly distributed throughout the volume and that
d
, the dimension
(
R
)
radius
=
r
radius
=
r
/
b
radius
=
r
/
b
2
Figure 2.15.
Here we schematically represent how a given mass can be non-uniformly distributed within a
given volume in such a way that the volume occupied by the mass has a fractal dimension given
by (
2.31
). The parameter
b
gives the scaling from the original sphere of radius
R
and the
parameter
a
gives the scaling from the original total mass
M
. Adapted from [
93
].
