PLoS supp. material by Hermann Cuntz
Therefore, the derivation of some elementary branching properties
follows directly from the graph representation of the tree.
As such, the child nodes of each node i can be read out in the non-zero
elements directly from dA in column i.
The index of the direct parent node idpar to any node i
is simply the i-th element of:
idpar = dA x (1 2 ... N)T
Further order r parents are simply obtained by applying repeated
iparr = dAr x (1 2 ... N)T
Where r = 0 corresponds to the node itself, r = 1 the parent,
r = 2 the grand-parent etc…
Correspondingly, the vector of topological path lengths PL
from all nodes to the root of the directed graph can be obtained as follows:
where dA1 is the first column of dA.
A similar approach can be used to obtain the vector of branch order values
for all elements compared to the root of the graph in position 1.
A supporting adjacency matrix sdA is required,
which is weighted by the number of child nodes of each node:
sdA = dA · (diag(sum(dA)))
By multiplying this matrix, branch points get potentiated, and the branch order BO (see "BO_tree") can be extracted by taking the base 2 logarithm:
where sdA1 is the first column of sdA.
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