![](bg5.png)
Figure 4: The x-axis represents the values of the causal-
ity between nodes divided in four sets. The y-axis repre-
sents the membership degree of each value in the x-axis for
each fuzzy set. In this manner, 0.25 belongs to the set of
some causality with a membership degree µ(0.25) = 1 rep-
resented in the y-axis. The value 0.75 belongs to the set
sufficient with a membership degree of 1 and so on.
a
1
= [0, 1, 0, 0.75, 0]
a
2
= [0, 0, 1, 0, 0.5]
a
3
= [0.25, 0.25, 0.25, 0, 0.25]
a
4
= [0.03, 0.28, 0.31, 0.18 0.15 ]
a
5
= [0.07, 0.10, 0.41, 0.02 0.21]
a
6
= [0.10, 0.18, 0.17, 0.05 0.15]
a
7
= [0.04, 0.14, 0.25, 0.07 0.13]
.
.
.
a
20
= [0.01, 0.02, 0.04, 0.01 0.02]
Which means that if we force the system only
at t
0
the system tends to the equilibrium vector
a
eq
= [0, 0, 0, 0, 0]. In these case the matrix coef-
ficients represent the system damping to the initial
perturbation. The system behavior given this par-
ticular structure can be interpreted in terms of the
scenarios and mitigation actions, e.g. can the matrix
weights be modified by specific actions or strategies?
3.2 Fuzzy Weighted System
The matrix associated with the fuzzy system is shown
in figure 5.
Turning on the node C
1
we have (a
0
= [1 0 0 0 0
0 0 0 0 0 0 0]), and iterating, the system converges
in two steps into vector: a
∗
2
= [1.00, 1.00, 1.00, 1.00,
0.75, 0.37, 0.87, 1.00, 1.00, 0.75, 1.00, 0.5]
Here we have forced C
1
, i.e. at each iteration a[1] = 1.
We found that the nodes 3, 5, 6, 7, 8, 10, and 12,
reach stable values, while the nodes 2, 4, 9, and 11,
have been limited to 1 by the threshold function.
This offers important information about the network.
As nodes 2, 4, 9, and 11 diverge, they could be
interpreted as sensitive nodes, i.e. these are the nodes
that can not reach an equilibrium with the present
conditions. Then, for example, is possible to think
about short-term actions for them in order to avoid
unwanted scenarios. Furthermore we can say that
even thought the forcing node is industrialization,
mitigation actions may be focused in more than one
node. When we restart C
1
only at t
0
the system is
damping again by the matrix coefficients.
a
1
= [0, 1.00, 0, 0, 0, 0, 0.5, 1.00, 1.00, 0,
0.75, 0.5]
a
2
= [0, 0.37, 1.00, 1.00, 0.75, 0.37, 0.37, 0, 0.75,
0.75, 0.37, 0]
In a
2
the node 4 has been thresholded form 1.18 to 1,
these is because this node has the greatest causality
in the network.
a
3
= [0, 0.53, 0.71, 1.00, 0.28, 0, 0, 0.09, 0, 0, 0.75,
0]
In a
3
the node 4 has been thresholded again.
a
4
= [0, 0.25, 0.78, 1.00, 0.39, 0, 0, 0, 0.07, 0.07,
0.60, 0]
In a
4
we also threshold node 4 from 1.07 to 1
a
5
= [0, 0.25, 0.5, 1, 0.18, 0, 0, 0, 0, 0, 0.64, 0]
In a
5
node 4 is threshold from 1.15 to 1
a
6
= [0, 0.25, 0.5, 0.75, 0.18, 0, 0, 0, 0, 0, 0.5, 0]
In step 6 node 4 need no threshold. From these point
the system converges to zero.
a
7
= [0, 0.18, 0.43, 0.71, 0.18, 0, 0, 0, 0, 0, 0.43, 0]
.
.
.
a
14
= [0, 0.08, 0.17, 0.29, 0.07, 0, 0, 0, 0, 0, 0.18, 0]
We can see how the subsystem with feedback re-
mains, but converges to zero as iterations progress.
The fact that the node 4 diverges give us important
information about the sensibility of this node to the
network conditions, i.e. node 4 has strong sensibility.
Another important analysis is what happens when
the network is at equilibrium and then the conditions
change? In order to analyze this we consider the
vector a
∗
2
that we obtained while maintaining forc-
ing. Then turn off the first node and calculate a∗M
sist
.
a
0
= [1.00, 1.00, 1.00, 1.00, 0.75, 0.37, 0.87,
1.00, 1.00, 0.75, 1.00, 0.5]
a
1
= [0, 1.90, 1.34, 3.18, 0.75, 0.37, 0.87, 1.09, 1.75,
0.75, 1.87, 0.5]
using threshold to limit the values of nodes 2, 3, 4, 8,
9, and 11
a
1
= [0, 1.00, 1.00, 1.00, 0.75, 0.37, 0.87, 1.00, 1.00,
0.75, 1.00, 0.5]
in the second iteration we have:
a
2
= [0, 0.90, 1.00, 1.00, 0.75, 0.37, 0.37, 0.09, 0.75,
0.75, 1.00, 0]
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