On the Convergence of Tanh Fuzzy General Gray Cognitive Maps

Fuzzy cognitive maps (FCMs) are widely used for modeling complex systems but are limited in handling uncertainty from imprecise or multi-interval data. To address this, extended models, such as fuzzy gray cognitive maps (FGCMs) and fuzzy general gray cognitive maps (FGGCMs) have been developed. In particular, FGGCMs can process general gray numbers, enabling more effective modeling of uncertainty. However, the convergence properties of FGGCMs remain underexplored, limiting their reliability in applications involving prediction, control, and decision-making. This article addresses this gap by establishing a rigorous theoretical framework for analyzing the convergence of FGGCMs using the tanh activation function. First, we define the metric and vector space structure of GGN and prove their completeness. Based on this, Banach’s fixed-point theorem is employed to derive sufficient conditions for the global convergence of FGGCMs to a unique fixed point. Moreover, convergence criteria are separately established for the kernel and grayness components. Finally, we show that existing convergence results for FCMs and FGCMs are special cases of the broader theorems proposed in this work. The main finding is the convergence theorems of tanh FGGCM and the results provide a solid mathematical foundation for future developments in learning algorithms and FCM-based modeling under uncertainty.