3.1. METRICS fOR BRANCHING NETWORKS
form a "parent" branch of order w according to the rule
where 8(wl> w 2 ) =1 if WI =W 2 and 0 otherwise. Note that in all of these
definitions we are moving from the outermost parts on the network inwards,
counter to the direction of growth. Thus, a parent branch of order w is
only produced when two offspring branches of order w - 1 come together.
In all cases where offspring branches of different orders join, the parent will
have the same order as the maximum of the two offspring . The pruning
method is more useful when a specimen is examined manually while the
algorithmic method is preferable when digitized data of a branching structure
are available.
Now that we have a way to break a network into logical pieces, we may
begin to analyze the way that these pieces fit together.
3.1.2 Horton Statistics
On a network with branch ordering, various natural quantities to measure arise. Some principal ones are n w , the number of branch segments
for a given order w; lw, the average branch segment length; a w , the average cross-sectional branch area; and the variation in these numbers from
order to order. In some cases, the volume or area surrounding a branch segment can be measured. For example, a w for river networks would refer to
the average area of land from which water drains into stream segments of
order w. Networks internal to bodies such as blood networks have a typical
volume surrounding each branch. For marine organisms, a similar volume
may be determined if a particular species "fills space;' i.e. its branches are
separated by some characteristic scale. However, for any tree-like organism,
the fact that there is no body surrounding the structure means that it may
grow "loosely" and such a measure would be irrelevant.
Horton (1945) and later Schumm (1956) observed for river networks that
network quantities such as those above approximately change by the same
ratio from order to order. For the present situation this would amount to the
follow statements:
69
n w = R n ,
n W+l
and
aW+l - R
- - - a ·
a w
(3-2)
All of these "Horton ratios" are defined so that they are greater than unity.
The numbers of branch segments decrease with order while branch segment
length and cross-sectional area usually increase . Typical values of the Horton
ratios are R; == 4, R, == 2 and Ra == 4.
It is to be expected that these relations hold over a range of orders but
not all orders and some care must be taken in measuring these Horton ratios .
Note that if the relations do not hold for a network then order-dependent
ratios should be used. So instead of a single value of R n we would have
Rn(w) = nw/n w-I •
Asan example, we can readily calculate the n w for the example network in
Fig.3.1.We find nl = 26, n« = 4, and n 3 =1. Note that for an order Q network,
we always have no = 1. Now, nJ n, = 4 and n3/n2 = 6.5 but for such a small
network we should not expect these to be the same. Care must be taken to
form a "parent" branch of order w according to the rule
where 8(wl> w 2 ) =1 if WI =W 2 and 0 otherwise. Note that in all of these
definitions we are moving from the outermost parts on the network inwards,
counter to the direction of growth. Thus, a parent branch of order w is
only produced when two offspring branches of order w - 1 come together.
In all cases where offspring branches of different orders join, the parent will
have the same order as the maximum of the two offspring . The pruning
method is more useful when a specimen is examined manually while the
algorithmic method is preferable when digitized data of a branching structure
are available.
Now that we have a way to break a network into logical pieces, we may
begin to analyze the way that these pieces fit together.
3.1.2 Horton Statistics
On a network with branch ordering, various natural quantities to measure arise. Some principal ones are n w , the number of branch segments
for a given order w; lw, the average branch segment length; a w , the average cross-sectional branch area; and the variation in these numbers from
order to order. In some cases, the volume or area surrounding a branch segment can be measured. For example, a w for river networks would refer to
the average area of land from which water drains into stream segments of
order w. Networks internal to bodies such as blood networks have a typical
volume surrounding each branch. For marine organisms, a similar volume
may be determined if a particular species "fills space;' i.e. its branches are
separated by some characteristic scale. However, for any tree-like organism,
the fact that there is no body surrounding the structure means that it may
grow "loosely" and such a measure would be irrelevant.
Horton (1945) and later Schumm (1956) observed for river networks that
network quantities such as those above approximately change by the same
ratio from order to order. For the present situation this would amount to the
follow statements:
69
n w = R n ,
n W+l
and
aW+l - R
- - - a ·
a w
(3-2)
All of these "Horton ratios" are defined so that they are greater than unity.
The numbers of branch segments decrease with order while branch segment
length and cross-sectional area usually increase . Typical values of the Horton
ratios are R; == 4, R, == 2 and Ra == 4.
It is to be expected that these relations hold over a range of orders but
not all orders and some care must be taken in measuring these Horton ratios .
Note that if the relations do not hold for a network then order-dependent
ratios should be used. So instead of a single value of R n we would have
Rn(w) = nw/n w-I •
Asan example, we can readily calculate the n w for the example network in
Fig.3.1.We find nl = 26, n« = 4, and n 3 =1. Note that for an order Q network,
we always have no = 1. Now, nJ n, = 4 and n3/n2 = 6.5 but for such a small
network we should not expect these to be the same. Care must be taken to