Appendix A. Proof of the Convergence of Matrix S
Let
be the matrix of absolute values, i.e.,
, where
is defined by Equation (
4). Thanks to the definition of scaling factor
s in Equation (
5), at least one row sum of elements of
is less than 1. Consequently,
has the same properties as the sub-matrix of transient states of the matrix representing absorbing Markov chains ([
34], Chapter 11). The power
of this matrix converges to 0 when
. If we allow also negative elements in matrix
, the general form of an element of
has the form
As this multiple sum contains both positive and negative terms, it is obvious that
and we can conclude that the series
of the matrix that contains both positive and negative elements converges absolutely (see e.g., [
35], Chapter 4).
Appendix B. Illustrative Example of the Extended WINGS Procedure
In this appendix we placed an illustrative example to present step-by-step the extended WINGS procedure. It also explains how the WINGS output can be interpreted and how it can be useful in problem analysis and decision making.
A subject of the example is a small abstract system which contains only six concepts. There are some positive and negative causal relations among concepts. The WINGS network (digraph) of the system is presented in
Figure A1. From the first view it is clear that node F plays central role in the system. It is an ultimate effect, as it does not influence any other node.
Figure A1.
WINGS network for the illustrative example.
Figure A1.
WINGS network for the illustrative example.
As we consider here an abstract case, we skip the beginning stage of the procedure–qualitative analysis—which is described in
Section 3.2.1 and proceed directly to quantitative evaluation (
Section 3.2.2). For both
internal strength and
influence we use the same nine point scale of integers 1–9. Node D is the strongest component of the system, while component E—the weakest. Though node F is not attributed with any internal strength. Putting its strength formally equal to zero does not distort the results of the WINGS procedure because node F does not influence any other node. In other words, the internal strength of F is irrelevant to the results of calculations.
Node D has the highest number (3) of direct impacts on other nodes, but one of them is negative. Node A exerts strongly on node F (value 8). Node E plays clearly negative role in the system. It has a direct negative impact on F, but also through paths of greater length (E-D-F with one negative segment, E-D-C-B-F with three negative segments). There is a single path (E-D-C-B-A-F with two negative segments) which represents positive influence of E on F. A few loops exist in a network, e.g., the Nodes A, B and C form a positive (reinforcing) loop, while D and E-a negative (balancing) one.
The next stage of the WINGS method (calculation of output described in
Section 3.2.2) begins with construction of direct strength-influence matrix
. The internal strength of each node is inserted on main diagonal of matrix
. The values of impacts are placed outside of main diagonal according to nodes indices. Then matrix
is scaled (see Equations (
4) and (
5)). Thanks to the relation between digraphs and matrix algebra the
p power of the matrix
contains the values of indirect impact of one node onto another calculated as the multiplication of impacts along the path of length
p. If the path of length
p does not exist, the corresponding element of
is equal zero.
Scaling used in the WINGS method causes that the values of impact decreases with the path length and ensures that the sum of all powers of matrix
converges (see
Appendix A). On the other hand, scaling leads also to small values in matrix
. (For the square matrix of size
representing the case study in this paper the order of the largest values is
. For this illustrative example it is
.) As the WINGS is based on ratio scale, to facilitate comparison of the results we can rescale the output measures. A convenient way is to rescale all values of Impact and Receptivity in relation to the largest of them, as we do it throughout this paper.
In
Table A1 the relative values of Impact, Receptivity, Involvement and Role are presented. Thanks to a limited number of nodes and uncomplicated structure of the system we can use visual examination of the network and intuition to verify the WINGS output. Node A has the strongest relative total impact followed by node D and C. Node B has relative total impact close to zero while node E—definitely negative. These results are in agreement with the intuitive view of the network. The high value of the total relative receptivity of node B may be justified by comparatively strong impact (5) from strong node C (5) which is a part of the reinforcing loop. Similarly, the second position of node E can be an effect of impact by the strong node D(7). A mix of the positive and negative impacts on node F leads to its average value of total relative receptivity.
Table A1.
The WINGS output for illustrative example—relative values.
Table A1.
The WINGS output for illustrative example—relative values.
| Impact | Receptivity | Involvement | Role |
---|
A | 1.0000 | 0.2065 | 1.2065 | 0.7935 |
B | 0.0700 | 0.6539 | 0.7239 | |
C | 0.7185 | 0.2767 | 0.9952 | 0.4418 |
D | 0.8747 | 0.2722 | 1.1469 | 0.6026 |
E | | 0.4242 | 0.8739 | 0.0256 |
F | 0.0000 | 0.3801 | 0.3801 | |
When we consider absolute total impact (
Table A2), we see that nodes A and D reversed their positions in comparison to relative ranking. If we look at the network it becomes completely understandable. Node D has higher internal strength than node A and its direct impacts are 6, 4 and 4 (absolute values), while node A has only two direct impacts with absolute values 8 and 3. Node C keeps its third position. Node E is classified higher than node B because now the negative signs of its two direct impacts are neglected. Similar effects can be observed in total absolute receptivity. Node F is the ‘strongest receiver’ with four incoming direct impacts. Node C, having two incoming direct impacts, takes the second position.
Table A2.
The WINGS output for illustrative example—absolute values.
Table A2.
The WINGS output for illustrative example—absolute values.
| Impact | Receptivity | Involvement | Role |
---|
A | 0.6369 | 0.1315 | 0.7684 | 0.5054 |
B | 0.3121 | 0.4516 | 0.7638 | |
C | 0.4811 | 0.5839 | 1.0650 | |
D | 1.0000 | 0.4638 | 1.4638 | 0.5362 |
E | 0.4617 | 0.2702 | 0.7318 | 0.1915 |
F | 0.0000 | 0.9908 | 0.9908 | |
In WINGS involvement and role are the most important measures in analysis of the system behaviour. A map showing locations of all system components on the plane with involvement and role as Cartesian co-ordinates is a convenient tool for such an analysis. In
Figure A2 we see a map of relative measures. Nodes A and D have the highest relative involvement among the system components. It means that they positively influence the system more than other components. Together with nodes C and E they form the influencing (cause) group (positive role). However, the role of node E is close to zero. The other nodes (B, and F) form the influenced (effect) group (negative role). Depending on the specific features of the studied system the values and relative locations of system components on the involvement-role plane help decision maker to evaluate the potential actions and choose the best one. Recognition of the roles in the system is a good example of value added by WINGS. Even for such simple example, as the one studied here, it would be difficult to recognise the roles of system components only by pure intuition.
Figure A2.
Relative Role vs. Relative Involvement of concepts in illustrative example.
Figure A2.
Relative Role vs. Relative Involvement of concepts in illustrative example.
Absolute involvement and absolute role bring an additional information that can be useful in system analysis (see
Figure A3). The absolute involvement of node D is stronger than that of node A. It means that negative part of node D impact is important in the system and node D deserves deeper analysis. For example questions of the kind: “Is it possible to weaken its negative influence and how to do it?” can be raised. The absolute receptivity of node C is higher than relative (opposite signs of influence from nodes A and D cause compensation) and, as a result, node C joins the influenced group in absolute measures.
Figure A3.
Absolute Role vs. Absolute Involvement of concepts in illustrative example.
Figure A3.
Absolute Role vs. Absolute Involvement of concepts in illustrative example.