Adaptive Estimation of Fuzzy Cognitive Networks and Applications

Abstract

Fuzzy Cognitive Networks (FCN) have been introduced by the authors as an operational extension of Fuzzy Cognitive Maps (FCM), which were initially in- troduced by Kosko to model complex behavioral systems in various scientific areas. One important issue of their operation 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 chapter, we present a study on the existence of solutions of FCMs equipped with continuous differentiable sigmoid functions having contractive or at least non-expansive prop- erties. This is done by using an appropriately defined contraction mapping theorem and the non-expansive mapping theorem. It is proved that when the weight inter- connections fulfill certain conditions the concept values will converge to a unique solution regardless the exact values of the initial concept values perturbations, or in some cases a solution exists that may not necessarily be unique. Otherwise the exist- ence or the uniqueness of equilibrium cannot be assured. Based on these results an adaptive weight estimation algorithm is proposed which employs appropriate weight projection criteria to assure that the uniqueness of FCM solution is not comprom- ised. Fuzzy Cognitive Networks are in the sequel invoked providing an application framework for the obtained results.