Agriculture Reference
In-Depth Information
positive if all positive-effect arcs leading to it had a positive impact and negative if
otherwise. The sign of a source vertex was the sum of the impacts of all arcs leading
from it. In contrast to a path, a
cycle
was defined as a sequence of vertices (
v
1
,
v
2
,
…,
v
t
) such that for all
i
= {1, 2, …,
t
} there is an arc (
v
i
,
v
(
i
+1)
), and where
v
1
=
v
t
,
while all other vertices are distinct. The sign, length, and impact of a cycle were as
defined for paths. The diagonal elements (
a
ii
) of the matrix
A
t
gave the number of
cycles and closed walks from a given vertex (
v
i
). The off-diagonal elements gave the
number of walks and paths from one vertex (
v
i
) to another (
v
j
). A walk was similar to
a path with the exception that the vertices forming the sequence were not distinct.
The total effect (
TE
) of a vertex (
v
i
) on another vertex (
v
j
) is the sum of the effects
of all the paths from
v
i
to
v
j
. If all such effects are positive, then the total effect is
positive (+); if all are negative, the total effect is negative (−); if two or more paths
of the same length have opposite effects, the sum is indeterminate (#), and if all the
paths with opposite effects are of different lengths, the sum is ambivalent (±). A
digraph with at least one indeterminate or ambivalent total effect is said to be path
imbalanced. One that has no indeterminate or ambivalent total effect is path bal-
anced. The signed adjacency matrix (also called the incidence matrix, direct effects
matrix, or valency matrix) is used to compute the total effect. The impact of vertex
v
i
on another vertex
v
j
is calculated as the total effect of
v
i
on
v
j
multiplied by the sign
of vertex
v
j
.
The reachability matrix
R
is a square
n
×
n
matrix with elements
r
ij
that are 1 if
v
j
is reachable from
v
i
and 0 if otherwise. By definition, each element is reachable
from itself, such that
r
ii
= 1 for all
i
. The reachability matrix can be computed from
the adjacency matrix using the formula
R
=
B
[(
I
+
A
)
n
−1
].
B
is a Boolean function
where
B
(
x
) = 0 if
x
= 0, and
B
(
x
) = 1 if
x
> 0.
I
is the identity matrix. The digraph
D
= (
V
,
A
) is said to be strongly connected (i.e., for every pair of vertices
v
i
and
v
j
,
v
i
is reachable from
v
j
and
v
j
is reachable from
v
i
) if and only if
R
=
J
, the matrix of all
1's.
D
is unilaterally connected (i.e., for every pair of vertices
v
i
and
v
j
,
v
i
is reachable
from
v
j
or
v
j
is reachable from
v
i
) if and only if
B
[
R
+
R
′] =
J
. The strong component
(i.e., a subdigraph of
D
where all the vertices are maximally connected) to which a
vertex (
v
i
) is a member is given by the entries of 1 in the
i
ith row (or column) of the
elementwise product of
R
and
R
′. The number of elements in each strong component
is given by the main diagonal elements of
R
2
.
4.2.3
p
u l s e
p
r o C e s s
m
o D e l s
A
weighted digraph
is one in which each arc (
v
i
,
v
j
) is associated with a weight (
a
ij
).
The signed adjacency matrix (in this case referred to as a weighted adjacency matrix)
of a weighted digraph therefore consists of the signed weights (
a
ij
) of all the arcs
(
v
i
,
v
j
) in the digraphs and is 0 if the arc does not exist. Under the pulse process, an
arc (
v
i
,
v
j
) was interpreted as implying that when the value of
v
i
is increased by one
unit at a discrete step
t
in time or space,
v
j
would increase (or decrease depending
on the sign of
a
ii
) by
a
ij
units at step
t
+ 1. Initially, the arcs in each digraph were
considered to be equal in weight and length. The models therefore assumed that a
pulse in vertex
v
i
at time
t
was related in a linear fashion to the pulse in
v
j
at time
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