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Fuzzy cognitive maps (FCMs) 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 of FCMs equipped with continuous differentiable sigmoid functions having contractive or, at least, non-expansive properties. This is done by using an appropriately defined contraction mapping theorem and the non-expansive mapping theorem. It is proved that when the weight interconnections fulfill certain conditions, the concept values will converge to a unique solution, regardless of the exact values of the initial concept values perturbations, or in some cases, a solution exists that may not necessarily be unique; otherwise, the existence or the uniqueness of equilibrium cannot be assured. Based on these results, an adaptive weight-estimation algorithm is proposed that employs appropriate weight projection criteria to assure that the uniqueness of FCM solution is not compromised. In view of these results, recently proposed extensions of FCM, which are the fuzzy cognitive networks (FCN), are invoked.
Date of Publication: Aug. 2009