Investigating the Convergence of Sigmoid-Based Fuzzy General Grey Cognitive Maps

TL;DR

This paper analyzes the convergence of Sigmoid-Based Fuzzy General Grey Cognitive Maps, deriving conditions for their unique fixed points and demonstrating their applicability in control, prediction, and decision support systems.

Contribution

It provides the first sufficient conditions for the convergence of FGGCM with sigmoid functions, extending existing theorems for FCM and FGCM.

Findings

Derived convergence conditions using fixed point theorems.

Validated that existing FCM and FGCM theorems are special cases.

Demonstrated applicability in control and decision systems.

Abstract

The Fuzzy General Grey Cognitive Map (FGGCM) and Fuzzy Grey Cognitive Map (FGCM) extend the Fuzzy Cognitive Map (FCM) by integrating uncertainty from multiple interval data or fuzzy numbers. Despite extensive studies on the convergence of FCM and FGCM, the convergence behavior of FGGCM under sigmoid activation functions remains underexplored. This paper addresses this gap by deriving sufficient conditions for the convergence of FGGCM to a unique fixed point. Using the Banach and Browder-Gohde-Kirk fixed point theorems, and Cauchy-Schwarz inequality, the study establishes conditions for the kernels and greyness of FGGCM to converge to unique fixed points. A Web Experience FCM is adapted to design an FGGCM with weights modified to GGN. Comparisons with existing FCM and FGCM convergence theorems confirm that they are special cases of the theorems proposed here. The conclusions support the application of FGGCM in domains such as control, prediction, and decision support systems.