Agriculture Reference
In-Depth Information
t
+ 1 if there was an arc (
v
i
,
v
j
) in the digraph. The value (
v
it
) of vertex
v
i
at time
t
was
calculated as:
n
∑
o
vv P
=++
−
sgn( ,)
v vP
it
it
(
1
)
it
(
−
1
)
j
i
jt
(
−
1
)
j
=
1
0
is a vector of external pulses or change in vertices
v
1
,
v
2
, …,
v
n
at step (
t
− 1);
sgn(
v
i
,
v
j
) is the sign of arc (
v
i
,
v
j
);
P
j
(
t
−1)
is referred to as a pulse and is the
j
th element
of the pulse vector
P
at the (
t
− 1)th row.
P
jt
is given by the difference
v
jt
−
v
j
(
t
−1)
for
t
> 0 and 0 otherwise.
A pulse process of a signed digraph
D
was defined by a vector of the starting
values at each vertex given by
V
s
= {
v
1
s
,
v
2
s
, …,
v
ns
} and a vector of the initial pulses
at each of the vertices given by
P
0
0
=
P
0
= {
P
10
,
P
20
, …,
P
n
0
}. Thus, the value at vertex
v
i
at step
t
= 0 was calculated as
u
i
0
=
u
is
+
p
i
0
.
A pulse process is autonomous if
pt
i
P
it
(
−1
)
0
() = 0 for all
t
> 0, that is, no other external
pulses are applied after the initial pulse
P
0
at step
t
= 0. In an autonomous pulse pro-
cess in a digraph,
D
= (
V
,
A
),
P
t
= (
P
0
*
A
t
). Further, a pulse process starting at vertex
v
i
is described as simple if
P
0
has the
i
th
entry equal to 1 and all other entries equal
to 0; that is, the system receives the initial pulse from a single vertex. Under a simple
autonomous pulse process, a unit pulse is propagated through the system starting at
the initial vertex
v
i
. Under this process, the value of vertex
v
i
at time
t
is given by
n
∑
vv
= +
−
sgn(
vvP
,)
it
it
()
1
j
i
jt
()
−
1
j
=
1
From this, it can be shown that in a simple autonomous pulse process starting at
vertex
v
i
, the value at vertex
v
j
at step
t
is given by
u
j
(
t
) =
u
j
(0) +
e
ij
, where
e
ij
is the
i
,
j
th element of a matrix
T
= (
A
+
A
2
+ … +
A
t
), where
A
is the weighted adjacency
matrix.
The effect of a vertex
v
i
on another
v
j
was positive if all pulses at
v
j
resulting
from a simple autonomous pulse at
v
i
were always positive, ambivalent if they were
oscillating, and positive if they were always negative. The impact was calculated as
described in the geometric analysis.
Based on the work of Klee (1989), a digraph was described as stable, value (or
quasi-) stable, semistable, or unstable under a given pulse process. A digraph was
stable under a pulse process if the values at each vertex converged to the origin as
t
→ ∞. It was described as value stable if the values at each vertex were bounded,
that is, there were numbers
B
j
so that
•
v
jt
•
<
B
j
for all
j
and 0 ≤
t
≤ ∞. A digraph was
semistable if the values at each vertex changed at a polynomial rather than an expo-
nential rate. It was unstable if the converse was true. A digraph was described as
pulse stable under a pulse process if the pulses at each vertex were bounded for 0 ≤
t
≤ ∞, that is,
•
p
jt
•
<
B
j
for all
t
. Stability properties of a digraph are related to the
eigenvalues of the weighted adjacency matrix. A digraph was stable under all pulse
processes if and only if each eigenvalue had a negative real part (Klee, 1989). If all
nonzero eigenvalues of
A
were distinct and at most 1 in magnitude, then the digraph
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