Convergence Conditions for Sigmoid-Based Fuzzy General gray Cognitive Maps: A Theoretical Study

Fuzzy General gray Cognitive Maps (FGGCMs) extend Fuzzy Cognitive Maps (FCMs) and Fuzzy gray Cognitive Maps (FGCMs) by employing general gray numbers to handle uncertainty arising from multi-interval data. While the convergence properties of FCMs and FGCMs are well-established, the convergence behavior of FGGCMs, especially those employing the widely adopted sigmoid activation function, remains insufficiently explored. Investigating sigmoid-based FGGCMs is of critical importance, as the unique handling of kernel and grayness for general gray numbers under the sigmoid function leads to convergence criteria fundamentally different from those derived for the tanh function. This work addresses the gap by developing a theoretical framework to analyze the convergence of sigmoid FGGCMs. By applying the Banach fixed-point theorem, sufficient conditions are derived to ensure global convergence to a unique fixed point, with separate criteria established for the kernel and grayness components. It is found that the convergence of grayness is contingent upon the convergence of the kernel. This relationship contrasts with the grayness convergence behavior exhibited in tanh-based FGGCMs. The proposed theorems generalize and subsume existing convergence results for FCMs and FGCMs. The theoretical results are validated by generalizing a classic Web Experience model into FGCM and FGGCM, indicating that smaller sigmoid steepness parameters ensure system convergence to a fixed point, whereas larger values tend to induce limit cycles. This study establishes a foundational convergence theory to support the development of reliable learning algorithms and to enable the application of FGGCMs in prediction, control, and decision-support systems operating under uncertainty.