Theoretical Analysis of the State Space of Fuzzy Cognitive Maps using Shrink Functions

Abstract

We proposed definitions and theorems regarding Fuzzy Cognitive Maps (FCMs), which allow estimating bounds for the activation value of each neuron and analyzing the covering and proximity of feasible activation spaces. The main theoretical findings suggest that the state space of any FCM model equipped with transfer F - functions shrinks infinitely with no guarantee for the FCM to converge to a fixed point but to its limit state space. This result, in conjunction with the covering and proximity values of FCM-based models, helps to understand their poor performance when solving complex simulation problems.