Theoretical study of interval-valued fuzzy cognitive maps: inference mechanisms, convergence, and centrality measures

Jindong Feng; Zengtai Gong; Jindong Feng; Zengtai Gong

doi:10.3934/math.2025800

AIMS Mathematics

2025, Volume 10, Issue 8: 17954-17981. doi: 10.3934/math.2025800

Previous Article Next Article

Research article Topical Sections

Theoretical study of interval-valued fuzzy cognitive maps: inference mechanisms, convergence, and centrality measures

Jindong Feng ^1,2,
Zengtai Gong ^{1
,
,}

1.
College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, China
2.
College of Mathematics and Computer Science, Northwest Minzu University, Lanzhou 730030, China

Received: 21 April 2025 Revised: 21 July 2025 Accepted: 30 July 2025 Published: 07 August 2025
MSC : 03B52, 05C72

Fuzzy cognitive maps (FCMs) have garnered significant attention for modeling and analyzing complex systems, owing to their ability to effectively handle uncertainty and nonlinear relationships. Interval-valued fuzzy cognitive maps (IVFCMs), as an extension of FCMs, further enhance this capability by incorporating interval-valued fuzzy numbers to better represent system uncertainty and complexity. Despite their potential, most existing research on IVFCMs has focused on applications, with limited advancement in their theoretical foundations. This study established a comprehensive theoretical framework for IVFCMs, addressing critical gaps in their mathematical basis and dynamic behavior. The key contributions are as follows: (1) Formalization of the inference mechanism, including a rigorous definition of state updates using interval-valued fuzzy operations, and a proof that IVFCMs reduce to conventional FCMs as interval widths approach zero; (2) establishment of convergence guarantees by deriving sufficient stability conditions through spectral radius analysis and the Banach fixed-point theorem for sigmoid activation functions; (3) the proposal of novel centrality measures, introducing interval-valued degree and closeness centrality, supported by an optimized Dijkstra algorithm for efficient identification of key nodes; and (4) empirical validation through a time series prediction case study, demonstrating the model's superior performance in managing uncertainty and noise across different datasets. Overall, this work addressed fundamental theoretical challenges related to inference, convergence, and structural analysis in IVFCMs, while also demonstrating their practical utility in complex system modeling.
- interval operation,
- interval-valued fuzzy cognitive maps,
- complex systems modeling,
- inference mechanisms,
- convergence,
- centrality measurement
Citation: Jindong Feng, Zengtai Gong. Theoretical study of interval-valued fuzzy cognitive maps: inference mechanisms, convergence, and centrality measures[J]. AIMS Mathematics, 2025, 10(8): 17954-17981. doi: 10.3934/math.2025800

Related Papers:

Abstract

Fuzzy cognitive maps (FCMs) have garnered significant attention for modeling and analyzing complex systems, owing to their ability to effectively handle uncertainty and nonlinear relationships. Interval-valued fuzzy cognitive maps (IVFCMs), as an extension of FCMs, further enhance this capability by incorporating interval-valued fuzzy numbers to better represent system uncertainty and complexity. Despite their potential, most existing research on IVFCMs has focused on applications, with limited advancement in their theoretical foundations. This study established a comprehensive theoretical framework for IVFCMs, addressing critical gaps in their mathematical basis and dynamic behavior. The key contributions are as follows: (1) Formalization of the inference mechanism, including a rigorous definition of state updates using interval-valued fuzzy operations, and a proof that IVFCMs reduce to conventional FCMs as interval widths approach zero; (2) establishment of convergence guarantees by deriving sufficient stability conditions through spectral radius analysis and the Banach fixed-point theorem for sigmoid activation functions; (3) the proposal of novel centrality measures, introducing interval-valued degree and closeness centrality, supported by an optimized Dijkstra algorithm for efficient identification of key nodes; and (4) empirical validation through a time series prediction case study, demonstrating the model's superior performance in managing uncertainty and noise across different datasets. Overall, this work addressed fundamental theoretical challenges related to inference, convergence, and structural analysis in IVFCMs, while also demonstrating their practical utility in complex system modeling.

References

[1]	B. Kosko, Fuzzy cognitive maps, International Journal of Man-Machine Studies, 24 (1986), 65–75. https://doi.org/10.1016/S0020-7373(86)80040-2
[2]	E. Bakhtavar, M. Valipour, S. Yousefi, R. Sadiq, K. Hewage, Fuzzy cognitive maps in systems risk analysis: a comprehensive review, Complex Intell. Syst., 7 (2021), 621–637. https://doi.org/10.1007/s40747-020-00228-2 doi: 10.1007/s40747-020-00228-2
[3]	A. Mourhir, Scoping review of the potentials of fuzzy cognitive maps as a modeling approach for integrated environmental assessment and management, Environ. Modell. Softw., 135 (2021), 104891. https://doi.org/10.1016/j.envsoft.2020.104891 doi: 10.1016/j.envsoft.2020.104891
[4]	S. Habib, M. Akram, Medical decision support systems based on fuzzy cognitive maps, Int. J. Biomath., 12 (2019), 1950069. https://doi.org/10.1142/S1793524519500694 doi: 10.1142/S1793524519500694
[5]	C. Borrero-Domínguez, T. Escobar-Rodríguez, Decision support systems in crowdfunding: a fuzzy cognitive maps (FCM) approach, Decis. Support Syst., 173 (2023), 114000. https://doi.org/10.1016/j.dss.2023.114000 doi: 10.1016/j.dss.2023.114000
[6]	H. A. Choukolaei, S. E. Mirani, M. Hejazi, P. Ghasemi, A GIS-based crisis management using fuzzy cognitive mapping: PROMETHEE approach (a potential earthquake in Tehran), Soft Comput., in press. https://doi.org/10.1007/s00500-023-09260-z
[7]	H. A. Choukolaei, S. E. Mirani, P. Ghasemi, M. J. Rezaee, System dynamics simulation follow-up fuzzy cognitive map for investigating the effect of risks on relief in crisis management, Eng. Appl. Artif. Intel., 136 (2024), 109002. https://doi.org/10.1016/j.engappai.2024.109002 doi: 10.1016/j.engappai.2024.109002
[8]	O. Orang, P. C. de Lima e Silva, F. G. Guimarães, Time series forecasting using fuzzy cognitive maps: a survey, Artif. Intell. Rev., 56 (2023), 7733–7794. https://doi.org/10.1007/s10462-022-10319-w doi: 10.1007/s10462-022-10319-w
[9]	Y. Boutalis, T. L. Kottas, M. Christodoulou, Adaptive estimation of fuzzy cognitive maps with proven stability and parameter convergence, IEEE Trans. Fuzzy Syst., 17 (2009), 874–889. https://doi.org/10.1109/TFUZZ.2009.2017519 doi: 10.1109/TFUZZ.2009.2017519
[10]	T. Kottas, Y. Boutalis, M. Christodoulou, Bi-linear adaptive estimation of fuzzy cognitive networks, Appl. Soft Comput., 12 (2012), 3736–3756. https://doi.org/10.1016/j.asoc.2012.01.025 doi: 10.1016/j.asoc.2012.01.025
[11]	C. J. K. Knight, D. J. B. Lloyd, A. S. Penn, Linear and sigmoidal fuzzy cognitive maps: an analysis of fixed points, Appl. Soft Comput., 15 (2014), 193–202. https://doi.org/10.1016/j.asoc.2013.10.030 doi: 10.1016/j.asoc.2013.10.030
[12]	G. Nápoles, E. Papageorgiou, R. Bello, K. Vanhoof, On the convergence of sigmoid fuzzy cognitive maps, Inform. Sciences, 349 (2016), 154–171. https://doi.org/10.1016/j.ins.2016.02.040 doi: 10.1016/j.ins.2016.02.040
[13]	C. Luo, X. Song, Y. J. Zheng, Algebraic dynamics of k-valued fuzzy cognitive maps and its stabilization, Knowl. Based Syst., 209 (2020), 106424. https://doi.org/10.1016/j.knosys.2020.106424 doi: 10.1016/j.knosys.2020.106424
[14]	I. Á. Harmati, M. F. Hatwágner, L. T. Kóczy, Global stability of fuzzy cognitive maps, Neural Comput. Appl., 35 (2023), 7283–7295. https://doi.org/10.1007/s00521-021-06742-9 doi: 10.1007/s00521-021-06742-9
[15]	L. C. Freeman, D. Roeder, R. R. Mulholland, Centrality in social networks: Ⅱ. experimental results, Soc. Networks, 2 (1979), 119–141. https://doi.org/10.1016/0378-8733(79)90002-9 doi: 10.1016/0378-8733(79)90002-9
[16]	K. Das, S. Samanta, M. Pal, Study on centrality measures in social networks: a survey, Soc. Netw. Anal. Min., 8 (2018), 13. https://doi.org/10.1007/s13278-018-0493-2 doi: 10.1007/s13278-018-0493-2
[17]	M. Jalili, A. Salehzadeh-Yazdi, S. Gupta, O. Wolkenhauer, M. Yaghmaie, O. Resendis-Antonio, et al., Evolution of centrality measurements for the detection of essential proteins in biological networks, Front. Physiol., 7 (2016), 375. https://doi.org/10.3389/fphys.2016.00375 doi: 10.3389/fphys.2016.00375
[18]	P. D. Meo, K. Musial-Gabrys, D. Rosaci, G. M. L. Sarnè, L. Aroyo, Using centrality measures to predict helpfulness-based reputation in trust networks, ACM Trans. Internet Techn., 17 (2017), 8. https://doi.org/10.1145/2981545 doi: 10.1145/2981545
[19]	R.-J. Hu, Q. Li, G.-Y. Zhang, W.-C. Ma, Centrality measures in directed fuzzy social networks, Fuzzy Inf. Eng., 7 (2015), 115–128. https://doi.org/10.1016/j.fiae.2015.03.008 doi: 10.1016/j.fiae.2015.03.008
[20]	Z. Lu, Q. Zhang, X. Du, D. Wu, F. Gao, A fuzzy social network centrality analysis model for interpersonal spatial relations, Knowl. Based Syst., 105 (2016), 206–213. https://doi.org/10.1016/j.knosys.2016.05.020 doi: 10.1016/j.knosys.2016.05.020
[21]	M. Obiedat, S. Samarasinghe, A novel semi-quantitative Fuzzy Cognitive Map model for complex systems for addressing challenging participatory real life problems, Appl. Soft Comput., 48 (2016), 91–110. https://doi.org/10.1016/j.asoc.2016.06.001 doi: 10.1016/j.asoc.2016.06.001
[22]	Q. Wang, Z. T. Gong, Structural centrality in fuzzy social networks based on fuzzy hypergraph theory, Comput. Math. Organ. Theory, 26 (2020), 236–254. https://doi.org/10.1007/s10588-020-09312-x doi: 10.1007/s10588-020-09312-x
[23]	W. Stach, L. Kurgan, W. Pedrycz, Higher-order fuzzy cognitive maps, In: NAFIPS 2006 - 2006 Annual meeting of the North American fuzzy information processing society, Montreal, QC, Canada, 03–06 June 2006,166–171. https://doi.org/10.1109/NAFIPS.2006.365402
[24]	H. A. Mohammadi, S. Ghofrani, A. Nikseresht, Using empirical wavelet transform and high-order fuzzy cognitive maps for time series forecasting, Appl. Soft Comput., 135 (2023), 109990. https://doi.org/10.1016/j.asoc.2023.109990 doi: 10.1016/j.asoc.2023.109990
[25]	X. Li, Y. Ma, Q. Zhou, X. Zhang, Sparse large-scale high-order fuzzy cognitive maps guided by spearman correlation coefficient, Appl. Soft Comput., 167 (2024), 112253. https://doi.org/10.1016/j.asoc.2024.112253 doi: 10.1016/j.asoc.2024.112253
[26]	D. K. Iakovidis, E. Papageorgiou, Intuitionistic fuzzy cognitive maps for medical decision making, IEEE Trans. Inf. Technol. Biomed., 15 (2010), 100–107. https://doi.org/10.1109/TITB.2010.2093603 doi: 10.1109/TITB.2010.2093603
[27]	L. Sha, Y. Shao, Y. Li, A framework of fermatean fuzzy cognitive map and its extension based on Hamacher operation, Eng. Appl. Artif. Intel., 136 (2024), 108676. https://doi.org/10.1016/j.engappai.2024.108676 doi: 10.1016/j.engappai.2024.108676
[28]	P. Hajek, O. Prochazka, Interval-valued fuzzy cognitive maps for supporting business decisions, In: 2016 IEEE international conference on fuzzy systems (FUZZ-IEEE), Vancouver, BC, Canada, 24–29 July 2016,531–536. https://doi.org/10.1109/FUZZ-IEEE.2016.7737732
[29]	J. Aguilar, A dynamic fuzzy-cognitive-map approach based on random neural networks, International Journal of Computational Cognition, 1 (2003), 91–107.
[30]	T. Hickey, Q. Ju, M. H. Van Emden, Interval arithmetic: from principles to implementation, J. ACM, 48 (2001), 1038–1068. https://doi.org/10.1145/502102.502106 doi: 10.1145/502102.502106
[31]	G. Deschrijver, Arithmetic operators in interval-valued fuzzy set theory, Inform. Sciences, 177 (2007), 2906–2924. https://doi.org/10.1016/j.ins.2007.02.003 doi: 10.1016/j.ins.2007.02.003
[32]	A. Piegat, L. Dobryakova, A decomposition approach to type 2 interval arithmetic, Int. J. Appl. Math. Comput. Sci., 30 (2020), 185–201. https://doi.org/10.34768/amcs-2020-0015 doi: 10.34768/amcs-2020-0015
[33]	P. Breiding, K. Rose, S. Timme, Certifying zeros of polynomial systems using interval arithmetic, ACM Trans. Math. Softw., 49 (2023), 11. https://doi.org/10.1145/3580277 doi: 10.1145/3580277
[34]	M. B. Gorzałczany, A method of inference in approximate reasoning based on interval-valued fuzzy sets, Fuzzy Set Syst., 21 (1987), 1–17. https://doi.org/10.1016/0165-0114(87)90148-5 doi: 10.1016/0165-0114(87)90148-5
[35]	D. Shan, L. Wang, W. Lu, J. Chen, Convex optimization based high-order fuzzy cognitive map modeling and its application in time series predicting, IEEE Access, 12 (2024), 12683–12698. https://doi.org/10.1109/ACCESS.2024.3355194 doi: 10.1109/ACCESS.2024.3355194
[36]	A. Saeed, C. Li, M. Danish, S. Rubaiee, G. Tang, Z. Gan, Hybrid bidirectional LSTM model for short-term wind speed interval prediction, IEEE Access, 8 (2020), 182283–182294. https://doi.org/10.1109/ACCESS.2020.3027977 doi: 10.1109/ACCESS.2020.3027977

Reader Comments

Your name:*

Email:*
© 2025 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)