the FCM is balanced or imbalanced is given by the Balance Degree. Various types of Balance Degree exist for signed directed graphs. We proposed a new type ...
A New Balance Degree for Fuzzy Cognitive Maps Athanasios K. Tsadiras, Konstantinos G. Margaritis Department of Applied Informatics University of Macedonia Egnatias 156, P.O. Box 1591 54006 Thessaloniki Greece Phone: +30-31-891891, Fax: +30-31-844536 email:{tsadiras,kmarg}@macedonia.uom.gr
ABSTRACT: In this paper a new balance degree is proposed for better evaluation of the conflicts that exist in the graph of Fuzzy Cognitive Maps (FCMs). FCMs are fuzzy weighted directed graphs with feedback that create models that emulate the behaviour of complex decision processes using fuzzy causal relations. The existence of two causal relations of opposite sign between two nodes makes the graph of the FCM imbalanced. The degree to which the whole digraph of the FCM is balanced or imbalanced is given by the Balance Degree. Various types of Balance Degree exist for signed directed graphs. We proposed a new type of Balance degree that is suitable for FCMs. The new degree is evaluated by exhaustively searching all the paths that are created in the graph and gives an indication of the dynamical behaviour that we should expect from the FCM. This is important because the conclusions that we draw from the FCM are coming from the study of its dynamical behaviour. KEYWORDS: Fuzzy Cognitive Maps, Artificial Neural Networks, Graph Theory
1. INTRODUCTION TO FUZZY COGNITIVE MAPS Fuzzy Cognitive Maps have been introduced by Kosko [1], [2] based on Axelord's work on Cognitive Maps [3] and are considered a combination of fuzzy logic and artificial neural networks. FCMs create models as collections of concepts and the various causal relations that exist between these concepts[4]. The concepts are represented by nodes and the causal relationships by directed arcs between the nodes. Each arc is accompanied by a weight that defines the type of causal relation between the two nodes. The sign of the weight determines the positive or negative causal relation between the two concepts-nodes. An example of FCM concerning public health is given in figure 1. Two nodes/concepts are connected either by a direct arc or by a path. An arc between two concepts indicates a direct causal relation while a path between two nodes of the digraph indicates an indirect causal relation. The sign of the indirect causal relationship is positive if the path has an even number of negative direct causal relationships, while it is negative if the path has an odd number of negative direct causal relationships. The total effect that a concept has to another is given based on all the indirect causal relationships that exist from the one concept to the other and also by the direct causal relation that might exist. The following rules are used to evaluate the total effect: The total effect is positive if all the indirect causal relationships that exist from the one concept to the other are positive and also the direct causal relationship that might exist is also positive. The total effect is negative if all the indirect causal relationships that exist from the one concept to the other are negative and also the direct causal relationship that might exist is also negative. The total effect is unknown if all the indirect causal relationships that exist from the one concept to the other are positive and the direct causal relationship are not of the same sign. For example concept C1 (Number of people in the city) in figure 1 is connected to concept C7 (Bacteria) by two paths. The first is the C1 (+)→ C4 (+)→ C7 and indicates a positive indirect causal relationship. The second path is the C1 (+)→ C3 (+)→ C5 (-)→ C7 and indicates a negative indirect causal relationship. In this case we can not conclude if the total effect of concept C1 to C7 is positive or negative, or in other word if the increase of the population of the city
will increase or decrease the bacteria.
C1
C2 +0.1
Number of people in a city
Migration into city
+0.6
+0.7
Modernization
C3
+0.9 -0.3
C5
+0.9
C4 Carbage per area
C6
Sanitation facilities -0.9
Number of diseases per 1000 residents
-0.9 +0.8
+0.9
C7 Bacteria per area
Figure 1: An FCM concerning public health [5]
2. BALANCE DEGREE An FCM is imbalanced if we can find two paths between the same two nodes that create causal relations of different sign. In the opposite case the FCM is balanced. The term “balanced” digraph was introduced in Graph Theory by Heider[6] and was extended by Cartwright and Harary[7]. In an imbalanced FCM we can not determine the sign of the total effect of a concept to another. Nozicka et.al. [8] and also Nakamura et.al. [9] based on the idea that as the length of the path increase, the indirect casual relation becomes weaken, proposed that the total effect should have the sign of the shortest path between the two nodes. Eden [10] on the other hand, proposed that the sign of the total effect should be the sign of the most important path where the most important path is the one that passes through the most important nodes [10]. The degree to which the digraph of the FCM is balanced or imbalanced is given by the Balance Degree of the digraph. Various types of Balance Degrees have been proposed, each of them being suitable for certain types of problems. Harary [11] proposed the following balance degree â: â=p/t (1) where p is the number of positive semicycles of the digraph and t is the total number of semicycles of the digraph (semicycle is a cycle in the digraph which can be created by not take into consideration the direction of the arcs). The closer to 1 is the degree â, the closer to balance is the digraph. The Balance Degree as it is proposed above does not take into consideration the length of the semicycles (paths). If we assume that the indirect casual relation weakens as the length of the path increases then another Balance Degree can be used. Norman and Roberts [12,13] for this case proposed the following degree:
∑p â= ∑t
m
f (m)
m
f (m)
m
or
(2)
m
∑p â= ∑n
m
f ( m)
m
f ( m)
m
(3)
m
where p m is the number of positive semicycles in the graph that have length m , nm is the number of negative semicycles in the graph that have length m, t m is the total number of semicycles in the graph that have length m and f (m) a monotonously increasing function, for example f (m) = 1 / m , f (m) = 1 / m 2 or f (m) = 1 / 2 m
3. THE NEW BALANCE DEGREE We propose new Balance Degree r that checks strictly the balance of the digraph and is very useful for FCMs. This degree can take a value among the whole interval [0,1]. The closer r is to 0 the closer to a completely balanced digraph is the graph. To calculate this new degree we check the signs of all the indirect causal relationships that exists among each pair of concepts Ci and C j . Lets assume that exist pij positive paths from concepts Ci to concept C j and that
nij is the number of negative paths from concepts Ci to concept C j . Now there are two cases: Case 1: pij + nij =2k In this case the total number of paths between the two concepts is even. If i=j then we refer to cycles. The case of a completely balanced digraph (r=0) occurs when we have either p ij =0 and nij =2k or pij =2k and nij =0. The case of a completely imbalanced FCM (r=1) occurs when pij = nij =k. The fraction
min{ pij , nij }
gives a measurement of the k balance according the above. If we have k+x, x>0 positive(negative) paths and k-x negative(positive) paths then the above fraction gives (k-x)/k which is right measurement for the balance of the FCM. Case 2 : pij + nij =2k+1 In this case the total number of paths between the two concepts is odd. The case of a completely balanced digraph (r=0) occurs when we have either pij =0 and nij =2k+1 or pij =2k+1 and nij =0. The case of a completely imbalanced FCM (r=1) occurs when pij =k and nij =k+1 or when pij =k+1 and nij =k. The fraction
min{ pij , nij }
gives a measurement of k +1 the balance according the above. If we have k-x, x>0 positive(negative) paths and k+x+1 negative(positive) paths then the above fraction gives (k-x)/(k+1) which is right measurement for the balance of the FCM. The degrees that the two cases above propose are different to the denominator. We can unify the two cases into one using the following fraction min{ p ij , n ij } (4) p ij + n ij + 1 int 2 where function int() returns the integer part of a decimal. The above fraction returns the balance degree of the paths between the concepts Ci and C j . To find the Balance Degree for the whole digraph we should add all these fractions for all the pairs of concepts. This gives the following equation
min{ p ij , nij } min{ p ii , nii } 1 + (5) r= 2 p + nii + 1 n i j , j ≠i p ij + nij + 1 i int ii int 2 2 where n is the total number of concepts in the FCM. The first term adds the partial balance degree for each different pair of concepts, while the second adds the partial balance degrees of the cycles in the graph. The factor 1 2 normalises n 2 the sum over the n terms.
∑∑
∑
This new Balance Degree is useful because it provides us with a clear indication of the number of conflicts that exist in the digraph of the FCM. By that we can draw conclusions about the dynamical behaviour that we should expect from the FCM. This is important because the forecasting process of the FCMs requires the study of the dynamical behaviour of the FCM model.
4. SUMMARY - CONCLUSIONS A new Balance Degree is proposed for the weight directed graph of Fuzzy Cognitive Maps. The new degree is evaluated by exhaustively searching all the paths that are created in the graph. The degree returns a value close to 0 if the FCM is almost balanced or close to1 if the FCM is almost completely imbalanced. This degree provides us with information concerning the conflicts that exist in the graph and gives an indication of the dynamical behaviour that we should expect from the FCM. This is important because the conclusions that we draw from the FCM are coming from the study of its dynamical behaviour.
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