(3.6)
(3-7)
3.1. METRICS FOR BRANCHING NETWORKS
Now, it turns out that in theory RT == Rl (Peckham 1995, Dodds and
Rothmann 1999) so we can understand RT as a ratio of length scales. The
parameter TI > 0 is the average number of side branches of one order lower
than the branch they are attached to, typically on the order of 1.0-1.5. In
general, larger values of (T I ) correspond to wider structures while smaller
values are in keeping with structures with relatively thinner profiles.
It can be shown that Horton's laws of branch numbers and branch segment length follow from what we have called Tokunaga's law, (3.3). The
required observation is that n w , the number of order co branches, is related
to the number of higher order branches that have order co side branches by
Q
nw = 2nw+J + L (Tw'-w)nw'
W '=W+l
The extra 2n w+J accounts for those order co branches that are not side
branches but rather generators of order co + 1 branches. A difference equation for R; is obtained by dividing through by nW+J and writing n w/ n W + I = R;
and n w ' /nw+J = (Rn)w+J-w' . For an infinitely large network, the solution is
1/2
Rn = AT + [A r -2RT]
(3.5)
where AT = (2 + RT + TI )/ 2. For a finite network, an exact solution can still
be obtained but is more complicated (Tokunaga 1978, Dodds and Rothmann
1999).
We see that the Horton ratios (3.2), although indicative of the network
structure, do not give the full picture. One cannot picture a network using the
Horton ratios alone since we do not know which branch segments connect
with which. The network perhaps suggested by the Horton ratios is one where
all branch segments of order co join branch segments of order co + 1, a true
hierarchy. But this is misleading since branch segments of a certain order
have side branches of all lower orders.
Nevertheless, one can further argue that Horton's laws also lead to
Tokunaga's law (Dodds and Rothmann 1999) An invertible transformation
between the remaining pairs of parameters may be deduced to be (Tokunaga
1978)
1/2
Rn = 1/2(2 + RT+ TI) + [(2 + RT+ TIP - 8RT]
R, =RT
This equivalence between the two descriptions means we have a useful crosscheck between measurements. Note that (R n , Rl) = (4,2) matches exactly with
(Til RT) = (1,2). Of course, this sort of equivalence applies only to exactly
self-similar networks. In practice deviations from self-similarity occur and
Tokunaga's statistics then carry more raw information (Cui et al. 1999, Dodds
and Rothman in prep.) .
Wehave thus defined a simple method for quantifying network structure
in branch ordering. Based on this ordering scheme, the two measures ofHorton and Tokunaga provide a good basis for comparison of network structure.
It is important to keep in mind that for small networks, strict self-similarity
is unlikely and that tables of numbers rather than poorly estimated ratios
are more useful (e.g. it would be wise to keep all of the n w along with any
estimate of Rn ) . Finally, wherever it is possible, statistics should always be
improved by averaging over many samples of a species.
71
(3-7)
3.1. METRICS FOR BRANCHING NETWORKS
Now, it turns out that in theory RT == Rl (Peckham 1995, Dodds and
Rothmann 1999) so we can understand RT as a ratio of length scales. The
parameter TI > 0 is the average number of side branches of one order lower
than the branch they are attached to, typically on the order of 1.0-1.5. In
general, larger values of (T I ) correspond to wider structures while smaller
values are in keeping with structures with relatively thinner profiles.
It can be shown that Horton's laws of branch numbers and branch segment length follow from what we have called Tokunaga's law, (3.3). The
required observation is that n w , the number of order co branches, is related
to the number of higher order branches that have order co side branches by
Q
nw = 2nw+J + L (Tw'-w)nw'
W '=W+l
The extra 2n w+J accounts for those order co branches that are not side
branches but rather generators of order co + 1 branches. A difference equation for R; is obtained by dividing through by nW+J and writing n w/ n W + I = R;
and n w ' /nw+J = (Rn)w+J-w' . For an infinitely large network, the solution is
1/2
Rn = AT + [A r -2RT]
(3.5)
where AT = (2 + RT + TI )/ 2. For a finite network, an exact solution can still
be obtained but is more complicated (Tokunaga 1978, Dodds and Rothmann
1999).
We see that the Horton ratios (3.2), although indicative of the network
structure, do not give the full picture. One cannot picture a network using the
Horton ratios alone since we do not know which branch segments connect
with which. The network perhaps suggested by the Horton ratios is one where
all branch segments of order co join branch segments of order co + 1, a true
hierarchy. But this is misleading since branch segments of a certain order
have side branches of all lower orders.
Nevertheless, one can further argue that Horton's laws also lead to
Tokunaga's law (Dodds and Rothmann 1999) An invertible transformation
between the remaining pairs of parameters may be deduced to be (Tokunaga
1978)
1/2
Rn = 1/2(2 + RT+ TI) + [(2 + RT+ TIP - 8RT]
R, =RT
This equivalence between the two descriptions means we have a useful crosscheck between measurements. Note that (R n , Rl) = (4,2) matches exactly with
(Til RT) = (1,2). Of course, this sort of equivalence applies only to exactly
self-similar networks. In practice deviations from self-similarity occur and
Tokunaga's statistics then carry more raw information (Cui et al. 1999, Dodds
and Rothman in prep.) .
Wehave thus defined a simple method for quantifying network structure
in branch ordering. Based on this ordering scheme, the two measures ofHorton and Tokunaga provide a good basis for comparison of network structure.
It is important to keep in mind that for small networks, strict self-similarity
is unlikely and that tables of numbers rather than poorly estimated ratios
are more useful (e.g. it would be wise to keep all of the n w along with any
estimate of Rn ) . Finally, wherever it is possible, statistics should always be
improved by averaging over many samples of a species.
71