On the Existence and Uniqueness of Fixed Points of Fuzzy Set Valued Sigmoid Fuzzy Cognitive Maps

Fuzzy cognitive maps are decision support tools, where the complex structure is modelled by a weighted, directed graph. The nodes represent specific characteristics of the modelled system, weighted and directed edges correspond to the direction and the strength of the relationship between the factors. The system state is identified by the values of the nodes which are computed by iteration. This process may lead to a fixed point, a limit cycle or produces chaotic behaviour. The type of behaviour depends on the weights, on the topology of the graph and on the function applied for the iteration. From the practical viewpoint, it is critical to know whether the iteration converges to a fixed point or not. In this article, we discuss this problem for the case when the weights or the values of the nodes are fuzzy numbers. This scenario may occur when linguistic variables, modelled by fuzzy numbers, describe the connections.