On the Existence and Uniqueness of Fixed Points of Fuzzy Cognitive Maps

Fuzzy Cognitive Maps (FCMs) are decision support tools, which were introduced to model complex behavioral systems. The final conclusion (output of the system) relies on the assumption that the system reaches an equilibrium point (fixed point) after a certain number of iteration. It is not straightforward that the iteration leads to a fixed point, since limit cycles and chaotic behaviour may also occur. In this article, we give sufficient conditions for the existence and uniqueness of the fixed point for log-sigmoid and hyperbolic tangent FCMs, based on the weighted connections between the concepts and the parameter of the threshold function. Moreover, in a special case, when all of the weights are non-negative, we prove that fixed point always exists, regardless of the parameter of the threshold function.