On the existence and uniqueness of solutions for the concept values in Fuzzy Cognitive Maps

Fuzzy cognitive maps (FCM) have been introduced by Kosko to model complex behavioral systems in various scientific areas. One issue that has not been adequately studied so far is the conditions under which they reach a certain equilibrium point after an initial perturbation. This is equivalent to studying the existence and uniqueness of solutions for their concept values. In this paper, we study the existence of solutions by using an appropriately defined contraction mapping theorem. It is proved that when the weight interconnections fulfill certain conditions the concept values will converge to a unique solution regardless the exact values of the initial concept values perturbations. Otherwise the existence or the uniqueness of equilibrium can not be assured. The results are considered very significant because set the basis for the development of reliable system identification and control schemes based on the concept of FCM. In view of these results recently proposed extensions of FCM, the fuzzy cognitive networks are invoked.