Webs, trees and branches - Complex Webs: Anticipating the Improbable - page 53

The growth rate is positive for 1 so that each successive generation is larger than

the one preceding, implying exponential growth to infinity. The growth rate is negative
for 0

λ>

1 so that each successive generation is smaller than the preceding one,
implying exponential decay to zero.
In da Vinci's tree it is easy to assign a generation number to each of the limbs, but
the counting procedure can become quite complicated in more complex networks like
the generations of the bronchial tubes in the mammalian lung, the blood vessels in
a capillary bed or the dendritic trees in a neuron. One form taken by the branching
laws is determined by the ratio of the average diameters of the tubes from one gen-
eration to the next, which we use subsequently. But for now we introduce the general
ratio

<λ<

z n
z n + 1 .

R z = (2.6)

In a geophysical context ( 2.6 ) is known as Horton's law for river trees and fluvial land-
scapes, where R z can be the ratio of the number of branches in successive generations
of branches, the average length of the river branch or the average area of the river seg-
ment at generation n . In any of these cases the parameter R z determines a branching
law and this equation implies a geometric self-similarity in the tree, as anticipated by
Thompson.
Dodds and Rothman [ 17 ] view the structure of river webs as discrete sets of nested
sub-webs built out of individual stream segments. In Figure 2.8 an example of stream
ordering for the Mississippi basin is given. In the lower graphic is a satellite picture
of the Mississippi basin and in the upper is a mathematical schematic representation
of its branching. How this schematic representation of the real world is constructed is
explained as follows [ 17 ]:

A source stream is defined as a section of stream that runs from a channel head to a junction with
another stream ... These source streams are classified as the first order stream segments of the
network. Next, remove these source streams and identify the new source streams of the remaining
network. These are the second order stream segments. The process is repeated until one stream
segment is left of order . The order of the network is then defined to be .

Note that Horton's law [ 31 ] is consistent with the simple scaling modeling adopted in
Figure 2.8 , but in addition requires a rather sophisticated method of counting to insure
that the stream quantities are properly indexed by the stream-branch ordering. It should
be emphasized that the ratios given are in terms of average quantities at sequential orders
and consequently the observation that the ratios in ( 2.6 ) are constant requires the proba-
bility distributions associated with these geomorphological quantities to have interesting
properties that we take up subsequently. Figure 2.9 depicts the empirical distribution of
stream lengths for the Mississippi River where a straight line on bi-logarithmic graph
paper would indicate that the distribution is an inverse power law beyond a certain
point.
Here we demonstrate the first connection between the spatial complexity depicted in
Figure 2.8 and an inverse power law characterizing some metric of the complex web. In
Figure 2.9 the metric is the main stream length of the Mississippi River. However, we