70
3. MEASURING GROWTH AND FORM
find ranges of orders where ratios are relatively constant. Typically,networks
of at least six orders are necessary for this.
Another quantity we can use is the distance from the end of an order w
sub-network out along branches to the furthest tip. Writing this length as lw
we see that it can be broken down into branch segments and we have that
-
w -
lw = Lk=1 li, A simple calculation shows that the corresponding ratio R{
must be equivalent to the ratio for branch segment lengths R{ (Dodds and
Rothmann 1999). We therefore have in principle three independent Horton
ratios R n , R[, and R a • For networks that are constrained to two dimensions
and fill space (e.g. river networks), only two of these ratios are independent
since Rn == Ra (Dodds and Rothmann 1999).An important point to remember
is that the relations in (3.2) are only for mean values of branch length and
cross-sectional area. In general, similar relationships hold for higher order
moments (Peckham and Gupta 1999, Dodds and Rothman in prep.). The
above should be seen as a first measurement after which a more careful
examination of frequency distributions of quantities such as lw and a w may
be carried out.
3.1.3 Tokunaga Statistics
Not surprisingly, branch ordering allows for other interesting metrics and
eventually Tokunaga introduced the idea of measuring side branch statistics
(Tokunaga 1966, 1978, 1984). As with Horton's work, the first setting for
Tokunaga's method was river networks. This technique arguably provides the
most useful measurement based on branch ordering but has only recently
received much attention (Turcotte et al. 1998, Dodds and Rothmann 1999).
The idea is simply, for a given network, to count the average number of
order v "side branches" attached to order J1 branch segments. This gives
(Tp,v), a set of double-indexed parameters for a network which we will call
"Tokunaga ratios:' Specifically, only side branches that attach along a branch
segment and not at its beginning or end are counted. Note that O ~ J1 > v ~ 1,
so we can view the Tokunaga ratios as a lower (or upper) triangular matrix
of numbers.
Referring back to Fig.3.1, we have three Tokunaga ratios to measure: T2,h
T 3 ,1 and T 3 ,2 ' From Fig.3.1b we see that there are two W = 2 side branches for
the single w =3 branch segment. Note that the other two w =2 branches
are not side branches since they meet the w = 3 branch segment at one end.
Therefore, we have TJ,2 = 2. It is left as an exercise for the reader to check that
T2,1 = 2.25 and T 3 ,1 = 9.
Tokunaga made several key observations about these side branch ratios
and while this was done in the context of river networks, the same notions
hold in general. The first is that because of the self-similar nature of branching
networks, the (Tp,v)should not depend absolutely on either of J1 or v but only
on the relative difference, i.e. k = J1 - v, The second, which also follows from
considerations of self-similarity, is that in changing the value of k = J1 - v,
the (Tp,v) must themselves change by a systematic ratio. These statements
lead to "Tokunaga's law":
Thus, for a strictly self-similar network only two parameters are necessary
to characterize the set of Tp,v: T 1 and RT.
3. MEASURING GROWTH AND FORM
find ranges of orders where ratios are relatively constant. Typically,networks
of at least six orders are necessary for this.
Another quantity we can use is the distance from the end of an order w
sub-network out along branches to the furthest tip. Writing this length as lw
we see that it can be broken down into branch segments and we have that
-
w -
lw = Lk=1 li, A simple calculation shows that the corresponding ratio R{
must be equivalent to the ratio for branch segment lengths R{ (Dodds and
Rothmann 1999). We therefore have in principle three independent Horton
ratios R n , R[, and R a • For networks that are constrained to two dimensions
and fill space (e.g. river networks), only two of these ratios are independent
since Rn == Ra (Dodds and Rothmann 1999).An important point to remember
is that the relations in (3.2) are only for mean values of branch length and
cross-sectional area. In general, similar relationships hold for higher order
moments (Peckham and Gupta 1999, Dodds and Rothman in prep.). The
above should be seen as a first measurement after which a more careful
examination of frequency distributions of quantities such as lw and a w may
be carried out.
3.1.3 Tokunaga Statistics
Not surprisingly, branch ordering allows for other interesting metrics and
eventually Tokunaga introduced the idea of measuring side branch statistics
(Tokunaga 1966, 1978, 1984). As with Horton's work, the first setting for
Tokunaga's method was river networks. This technique arguably provides the
most useful measurement based on branch ordering but has only recently
received much attention (Turcotte et al. 1998, Dodds and Rothmann 1999).
The idea is simply, for a given network, to count the average number of
order v "side branches" attached to order J1 branch segments. This gives
(Tp,v), a set of double-indexed parameters for a network which we will call
"Tokunaga ratios:' Specifically, only side branches that attach along a branch
segment and not at its beginning or end are counted. Note that O ~ J1 > v ~ 1,
so we can view the Tokunaga ratios as a lower (or upper) triangular matrix
of numbers.
Referring back to Fig.3.1, we have three Tokunaga ratios to measure: T2,h
T 3 ,1 and T 3 ,2 ' From Fig.3.1b we see that there are two W = 2 side branches for
the single w =3 branch segment. Note that the other two w =2 branches
are not side branches since they meet the w = 3 branch segment at one end.
Therefore, we have TJ,2 = 2. It is left as an exercise for the reader to check that
T2,1 = 2.25 and T 3 ,1 = 9.
Tokunaga made several key observations about these side branch ratios
and while this was done in the context of river networks, the same notions
hold in general. The first is that because of the self-similar nature of branching
networks, the (Tp,v)should not depend absolutely on either of J1 or v but only
on the relative difference, i.e. k = J1 - v, The second, which also follows from
considerations of self-similarity, is that in changing the value of k = J1 - v,
the (Tp,v) must themselves change by a systematic ratio. These statements
lead to "Tokunaga's law":
Thus, for a strictly self-similar network only two parameters are necessary
to characterize the set of Tp,v: T 1 and RT.