Dynamics of Fuzzy Cognitive Maps with Uncertain Weights

In real-life problems, the weights of the connections between the concepts are often not known exactly. This uncertainty can be modelled in different ways, based on different considerations. In this chapter, we discuss two models, regarding the nature of uncertainty. Fuzzy grey cognitive maps (FGCMs) employ grey system theory to represent vagueness of the causal connections. This approach assumes that we do know the range of the unknown weight, but within this range we do not have any further information. In other words, the model assumes known extension, but unknown intension. In classical fuzzy cognitive maps the weights are single numbers, so the fuzziness is not present there. Fuzzy set valued fuzzy cognitve maps handle this known issue by assigning fuzzy sets to the connections, so in this model the intension of a weight is known, but the extension is not exactly clear. These models behave somewhat similarly to classical fuzzy cognitive maps. The final conclusion about the modelled system is based on the limiting behaviour of the iteration process. This iteration may converge to an equilibrium point, may arrive to limit cycles or may produce unstable behaviour. In this chapter, sufficient conditions for the existence and uniqueness of fixed points of fuzzy grey and fuzzy set valued FCMs are provided.