Stability of fuzzy cognitive maps with interval weights

Abstract

In fuzzy cognitive maps (FCMs) based modelling paradigm, the complex system's behaviour is gathered by the causal connections acting between its main characteristics or subsystems. The system is represented by a weighted, directed digraph, where the nodes represent specific subsystems or features, while the weighted and directed edges express the direction and strength of causal relations between them. The state of the complex system represented by the so-called activation values of the nodes, that is computed by an iterative method. The FCM based decision-making relies on the assumption that this iteration reaches an equilibrium point (fixed point), but other types of behaviour, namely limit cycles and chaotic patterns may also show up. In practice, the weights of connections are estimated by human experts or machine learning methods. Both cases have their own uncertainty, which can be represented by using intervals as weights instead of crisp numbers. In this paper, sufficient conditions are provided for the existence and uniqueness of fixed points of fuzzy cognitive maps that are equipped with interval weights, which also ensure the global asymptotic stability of the system.