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

Xudong Gao, Xiaoguang Gao, Jia Rong, Xiaolei Li, Ni Li, Yifeng Niu, Jun Chen

TL;DR

The paper addresses convergence analysis for sigmoid-based FGGCMs, a grey-number–aware extension of fuzzy cognitive maps. Using Banach and Browder-Gohde-Kirk fixed-point frameworks, it derives sufficient conditions for both kernel and greyness convergence to fixed points when using sigmoid activations with general grey numbers. Key contributions include unifying FCM, FGCM, and FGGCM under a common theoretical framework, enabling convergence guarantees even with complex GGNs and mixed-sign weights, and demonstrating practical validation on Web Experience scenarios. The findings enhance the reliability and applicability of FGGCMs in control, prediction, and decision-support tasks under uncertainty, while outlining future work on other activations and more efficient learning algorithms.

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.

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

TL;DR

The paper addresses convergence analysis for sigmoid-based FGGCMs, a grey-number–aware extension of fuzzy cognitive maps. Using Banach and Browder-Gohde-Kirk fixed-point frameworks, it derives sufficient conditions for both kernel and greyness convergence to fixed points when using sigmoid activations with general grey numbers. Key contributions include unifying FCM, FGCM, and FGGCM under a common theoretical framework, enabling convergence guarantees even with complex GGNs and mixed-sign weights, and demonstrating practical validation on Web Experience scenarios. The findings enhance the reliability and applicability of FGGCMs in control, prediction, and decision-support tasks under uncertainty, while outlining future work on other activations and more efficient learning algorithms.

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.

Paper Structure

This paper contains 8 sections, 9 theorems, 48 equations, 5 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Preparatory Knowledge
  3. The Convergence of sigmoid FGGCM
  4. Experiments
  5. Discussion
  6. The Convergence of Kernels
  7. The Convergence of Greyness
  8. Conclusions

Key Result

Theorem 1

In an FCM employing the sigmoid activation function $f(x) = \frac{1}{1 + \mathrm{e}^{-\lambda x}}$, a unique fixed point exists for each concept value $A_i$ if the following condition is satisfied: When the condition $\left\lVert \mathbf{W} \right\rVert _F = \frac{4}{\lambda}$ holds, at least one fixed point is guaranteed for each concept value $A_i$. The term $\left\lVert \mathbf{W} \right\rVert

Figures (5)

  • Figure 1: The Web Experience FCM constructed for users.
  • Figure 2: The output results of the Web Experience FCM across varying $\lambda$ values.
  • Figure 3: The results generated by the Web Experience FGCM for various values of $\lambda$.
  • Figure 4: The results produced by the Web Experience FGGCM across different values of $\lambda$.
  • Figure 8: The variation in greyness of the Web Experience FGGCM across different values of $\lambda$.

Theorems & Definitions (15)

  • Definition 1
  • Definition 2
  • Definition 3
  • Theorem 1
  • Theorem 2
  • Lemma 1: Banach Fixed Point Theorem
  • Lemma 2: Browder-Gohde-Kirk Fixed Point Theorem
  • Lemma 3
  • Proof 1
  • Theorem 3
  • ...and 5 more