Fuzzy cognitive map

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Rod Tabers FCM depicting eleven factors of the American drug market

A Fuzzy cognitive map is a cognitive map within which the relations between the elements (e.g. concepts, events, project resources) of a "mental landscape" can be used to compute the "strength of impact" of these elements. Ron Axelord introduced Cognitive Maps as a formal way of representing social scientific knowledge and modeling decision making in social and political systems. Then brought in the computation fuzzy logic.[1]


Fuzzy cognitive maps are signed fuzzy digraphs. They may look at first blush like Hasse diagrams but they are not. Spreadsheets or tables are used to map FCMs into matrices for further computation.[2][3][4] FCM is a technique used for causal knowledge acquisition and representation, it supports causal knowledge reasoning process and belong to the neuro-fuzzy system that aim at solving decision making problems, modeling and simulate complex systems. Learning algorithms have been proposed for training and updating FCMs weights mostly based on ideas coming from the field of Artificial Neural Networks. Adaptation and learning methodologies used to adapt the FCM model and adjust its weights. Kosko and Dickerson (Dickerson & Kosko, 1994) suggested the Differential Hebbian Learning (DHL) to train FCM. There have been proposed algorithms based on the initial Hebbian algorithm; others algorithms come from the field of genetic algorithms, swarm intelligence and evolutionary computation. Learning algorithms are used to overcome the shortcomings that the traditional FCM present i.e. decreasing the human intervention by suggested automated FCM candidates; or by activating only the most relevant concepts every execution time; or by making models more transparent and dynamic. .

Fuzzy cognitive maps (FCMs) have gained considerable research interest due to their ability in representing structured knowledge and model complex systems in various fields. This growing interest led to the need for enhancement and making more reliable models that can better represent real situations. A first simple application of FCMs is described in a book[5] of William R. Taylor, where the war in Afghanistan and Iraq is analyzed. And in Bart Kosko's book Fuzzy Thinking,[6] several Hasse diagrams illustrate the use of FCMs. As an example, one FCM quoted from Rod Taber[7] describes 11 factors of the American cocaine market and the relations between these factors. For computations, Taylor uses pentavalent logic (scalar values out of {-1,-0.5,0,+0.5,+1}). That particular map of Taber uses trivalent logic (scalar values out of {-1,0,+1}). Taber et al. also illustrate the dynamics of map fusion and give a theorem on the convergence of combination in a related article [8]

While applications in social sciences[5][6][7][9] introduced FCMs to the public, they are used in a much wider range of applications, which all have to deal with creating and using models[10] of uncertainty and complex processes and systems. Examples:

  • In business FCMs can be used for product planning.[11]
  • In economics, FCMs support the use of game theory in more complex settings.[12]
  • In Medical applications to model systems, provide diagnosis , develop decision support systems and medical assessment.
  • In Engineering for modeling and control mainly of complex systems
  • In project planning FCMs help to analyze the mutual dependencies between project resources.
  • In robotics[6][13] FCMs support machines to develop fuzzy models of their environments and to use these models to make crisp decisions.
  • In computer assisted learning FCMs enable computers to check whether students understand their lessons.[14]
  • In expert systems[7] a few or many FCMs can be aggregated into one FCM in order to process estimates of knowledgeable persons.[15]
  • In IT project management, a FCM-based methodology helps to success modelling.[16]

FCMappers[17] - an international online community for the analysis and the visualization of fuzzy cognitive maps offer support for starting with FCM and also provide a MS-Excel based tool that is able to check and analyse FCMs. The output is saved as Pajek file and can be visualized within 3rd party software like Pajek, Visone,... . They also offer to adapt the software to specific research needs. On their webpage you also will find a linklist for interesting scientific articles, related software, institutes, people and projects. The FCMappers have about one thousand registered members worldwide.

Additional FCM software tools, such as Mental Modeler,[18][19] have recently been developed as a decision-support tool for use in social science research, collaborative decision-making, and natural resource planning.

Bipolar Fuzzy Cognitive Maps[edit]

Fuzzy cognitive maps have been further extended to bipolar fuzzy cognitive maps based on bipolar fuzzy sets [20] and bipolar cognitive mapping. [21] [22] [23] [24] Bipolar fuzzy set theory as an equilibrium-based extension to fuzzy sets is recognized by L. A. Zadeh. [25]


  1. ^ Bart Kosko, Fuzzy Cognitive Maps, International Journal of Man-Machine Studies, 24(1986) 65-75 (first introduction of FCMs): [1] see also [2]
  2. ^ FCMapper - Excel based FCM analysis and visualization tool: http://www.FCMappers.net/joomla/index.php?option=com_content&view=article&id=52&Itemid=53
  3. ^ On line calculator and downloadable Java applications for FCM computations: http://www.ochoadeaspuru.com/fuzcogmap/index.php
  4. ^ Java standalone library for FCM computations: http://jfcm.megadix.it/
  5. ^ a b William R. Taylor: Lethal American Confusion[dead link] (How Bush and the Pacifists Each Failed in the War on Terrorism), 2006, ISBN 0-595-40655-6 (FCM application in chapter 14)
  6. ^ a b c Bart Kosko: Fuzzy Thinking, 1993/1995, ISBN 0-7868-8021-X (Chapter 12: Adaptive Fuzzy Systems)
  7. ^ a b c Rod Taber: Knowledge Processing with Fuzzy Cognitive Maps, Expert Systems with Applications, vol. 2, no. 1, 83-87, 1991 (Hasse diagram in German Wikipedia)
  8. ^ Rod Taber, Ronald R. Yager, and Cathy M. Helgason:Quantization Effects on the Equilibrium Behavior of Combined Fuzzy Cognitive Maps, International Journal of Intelligent Systems, vol. 22, 181-202, 2007.
  9. ^ Costas Neocleous, Christos Schizas, Costas Yenethlis: Fuzzy Cognitive Models in Studying Political Dynamics - The case of the Cyprus problem Archived September 29, 2007 at the Wayback Machine
  10. ^ Chrysostomos D. Stylios, Voula C. Georgopoulos, Peter P. Groumpos: The Use of Fuzzy Cognitive Maps in Modeling Systems[dead link]
  11. ^ Antonie Jetter: Produktplanung im Fuzzy Front End, 2005, ISBN 3-8350-0144-2
  12. ^ Vesa A. Niskanen: Application of Fuzzy Linguistic Cognitive Maps to Prisoner's Dilemma, 2005, ICIC International pp. 139-152, ISSN 1349-4198 Archived September 29, 2007 at the Wayback Machine
  13. ^ Marc Böhlen: More Robots in Cages,
  14. ^ Benjoe A. Juliano, Wylis Bandler: Tracing Chains-of-Thought (Fuzzy Methods in Cognitive Diagnosis), Physica-Verlag Heidelberg 1996, ISBN 3-7908-0922-5
  15. ^ W. B. Vasantha Kandasamy, Florentin Smarandache: Fuzzy Cognitive Maps and Neutrosophic Cognitive Maps, 2003, ISBN 1-931233-76-4
  16. ^ L. Rodriguez-Repiso, R. Setchi, and J.L. Salmeron. Modelling IT Projects success with Fuzzy Cognitive Maps. Expert Systems with Applications 32(2) pp. 543-559. 2007.
  17. ^ FCMappers - international community for fuzzy cognitive mapping: http://www.FCMappers.net/
  18. ^ Gray, S. Gray, S., Cox, L., and Henly-Shepard, S. 2013 Mental modeler: A fuzzy-logic cognitive mapping modeling tool for adaptive environmental management. Proceedings of the 46th International Conference on Complex Systems. 963-973. http://www.computer.org/csdl/proceedings/hicss/2013/4892/00/4892a965.pdf
  19. ^ http://www.mentalmodeler.com/
  20. ^ Wen-Ran Zhang, 1998, (Yin)(Yang) Bipolar Fuzzy Sets. Proceedings of IEEE World Congress on Computational Intelligence – Fuzz-IEEE, Anchorage, AK, 835-840
  21. ^ Wen-Ran Zhang, S. Chen & J. C. Bezdek, 1989, POOL2: A Generic System for Cognitive Map Development and Decision Analysis, IEEE Trans. on SMC., Vol. 19, No. 1, 1989, 31-39.
  22. ^ Wen-Ran Zhang, Chen, S., Wang, W. & King, R., 1992, A Cognitive Map Based Approach to the Coordination of Distributed Cooperative Agents. IEEE Trans. on SMC, Vol. 22, No. 1, 1992, 103-114.
  23. ^ Wen-Ran Zhang, 2003a, Equilibrium Relations and Bipolar Cognitive Mapping for Online Analytical Processing with Applications in International Relations and Strategic Decision Support. IEEE Trans. on SMC., Part B, Vol. 33. No. 2, 2003, 295-307.
  24. ^ Wen-Ran Zhang, 2003b, Equilibrium Energy and Stability Measures for Bipolar Decision and Global Regulation. Int’l J. of Fuzzy Sys. Vol. 5, No. 2, 2003, 114-122
  25. ^ L. A. Zadeh, 2008, Fuzzy logic. Scholarpedia, 3(3):1766, Created: 10 July 2006, reviewed: 27 March 2007, accepted: 31 March 2008.

External links[edit]

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