On the Existence and Uniqueness of Fixed Points of Fuzzy Cognitive Maps (original) (raw)

Abstract

Fuzzy Cognitive Maps (FCMs) are decision support tools, which were introduced to model complex behavioral systems. The final conclusion (output of the system) relies on the assumption that the system reaches an equilibrium point (fixed point) after a certain number of iteration. It is not straightforward that the iteration leads to a fixed point, since limit cycles and chaotic behaviour may also occur.

In this article, we give sufficient conditions for the existence and uniqueness of the fixed point for log-sigmoid and hyperbolic tangent FCMs, based on the weighted connections between the concepts and the parameter of the threshold function. Moreover, in a special case, when all of the weights are non-negative, we prove that fixed point always exists, regardless of the parameter of the threshold function.

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Acknowledgments

This work was supported by EFOP-3.6.2-16-2017-00015, HU-MATHS-IN – Intensification of the activity of the Hungarian Industrial Innovation Service Network and by National Research, Development and Innovation Office (NKFIH) K124055.

Author information

Authors and Affiliations

  1. Department of Mathematics and Computational Sciences, Széchenyi István University, Győr, Hungary
    István Á. Harmati
  2. Department of Information Technology, Széchenyi István University, Győr, Hungary
    Miklós F. Hatwágner & László T. Kóczy
  3. Department of Telecommunications and Mediainformatics, Budapest University of Technology and Economics, Budapest, Hungary
    László T. Kóczy

Authors

  1. István Á. Harmati
  2. Miklós F. Hatwágner
  3. László T. Kóczy

Corresponding author

Correspondence toIstván Á. Harmati .

Editor information

Editors and Affiliations

  1. Universidad de Cádiz, Cádiz, Cadiz, Spain
    Jesús Medina
  2. Universidad de Málaga, Málaga, Málaga, Spain
    Manuel Ojeda-Aciego
  3. Universidad de Granada, Granada, Spain
    José Luis Verdegay
  4. Universidad de Granada, Granada, Spain
    David A. Pelta
  5. Universidad de Málaga, Málaga, Málaga, Spain
    Inma P. Cabrera
  6. LIP6, Université Pierre et Marie Curie, CNRS, Paris, France
    Bernadette Bouchon-Meunier
  7. Iona College, New Rochelle, New York, USA
    Ronald R. Yager

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Harmati, I.Á., Hatwágner, M.F., Kóczy, L.T. (2018). On the Existence and Uniqueness of Fixed Points of Fuzzy Cognitive Maps. In: Medina, J., et al. Information Processing and Management of Uncertainty in Knowledge-Based Systems. Theory and Foundations. IPMU 2018. Communications in Computer and Information Science, vol 853. Springer, Cham. https://doi.org/10.1007/978-3-319-91473-2\_42

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