Unveiling the Dynamic Behavior of Fuzzy Cognitive Maps

Fuzzy Cognitive Maps (FCMs) are recurrent neural networks comprised of well-defined concepts and causal relations. While the literature about real-world FCM applications is prolific, the studies devoted to understanding the foundations behind these neural networks are rather scant. In this paper, we introduce several definitions and theorems that unveil the dynamic behavior of FCM-based models equipped with transfer F-functions. These analytical expressions 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 help understand their poor performance when solving complex simulation problems.