Abstract
Atanassov’s intuitionistic fuzzy set described the uncertainty of real-life events with the help of a membership and a non-membership degree. However, human opinion cannot be restricted to yes or no, but there is some abstinence and refusal degree as well. In such cases, picture fuzzy set is a suitable solution which described the abstinence and refusal grade of human opinion along with membership and non-membership grades. The aim of this paper is to analyse a social network and a wife network using the concept of picture fuzzy graph (PFG). For this purpose, the concept of PFG is proposed and some basic terms are demonstrated including complement, degree and bridges. The main advantage of the proposed PFG is that it describes the uncertainty in any real-life event with the help of four membership degrees where the traditional FG and IFG fail to be applied. The viability of PFG is shown by utilizing the concept in demonstrating two real-life problems including a social network and a Wi-Fi-network. A comparison of PFG with existing notions is established showing its superiority over the existing frameworks.
Similar content being viewed by others
References
Akram M, Davvaz B (2012) Strong intuitionistic fuzzy graphs. Filomat 26:177–196
Akram M, Dudek W (2013) A, Intuitionistic fuzzy hypergraphs with applications. Inf Sci 218:182–193
Atanassov KT (1999) Intuitionistic fuzzy sets. In: Intuitionistic fuzzy sets. Studies in fuzziness and soft computing, vol 35. Physica, Heidelberg. https://doi.org/10.1007/978-3-7908-1870-3_1
Atanassov K (2012) On intuitionistic fuzzy set theory. Springer, Heidelberg
Bezdek JC, Harris JD (1978) Fuzzy partitions and relations an axiomatic basis for clustering. Fuzzy Sets Syst 1:111–127
Bui CC (2013a) Picture fuzzy sets: first results. Part 1, In Preprint of Seminar on Neuro-Fuzzy Systems with Applications, Institute of Mathematics, Hanoi
Bui CC (2013b) Picture fuzzy sets: first results. Part 2, In Preprint of Seminar on Neuro-Fuzzy Systems with Applications, Institute of Mathematics, Hanoi
Bui CC, Vladik K (2013) Picture Fuzzy Sets- a new concept for computational intelligence problems. In: Proceedings of the third world congress on information and communication technologies WIICT, pp 1–6
Chen SM, Randyanto Y, Cheng SH (2016) Fuzzy queries processing based on intuitionistic fuzzy social relational networks. Inf Sci 327:110–124
Davvaz B, Jan N, Mahmood T, Ullah K (2019) Intuitionistic fuzzy graphs of n th type with applications. J Intell Fuzzy Syst 36(4):3923–3932
Ding B (1992) A clustering dynamic state method for maximal trees in fuzzy graph theory. J Numer Methods Comput Appl 13:157–160
Gani AN, Begum SS (2010) Degree, order and size in intuitionistic fuzzy graphs. Int J Algorithms Comput Math 3(3):11–16
Garg H (2017) Some picture fuzzy aggregation operators and their applications to multi-criteria decision-making. Arab J Sci Eng 42(12):5275–5290
Harary F (1959) Graph theoretic methods in the management sciences. Manag Sci 5:387–403
Harary F, Norman RZ (1953) Graph theory as a mathematical model in social science. Institute for Social Research, Ann Arbor
Harary F, Ross IC (1953) The number of complete cycles in a communication network. J Soc Psychol 40:329–332
Jan N, Ullah K, Mahmood T, Garg H, Davvaz B, Saeid AB, Broumi S (2019a) Some root level modifications in interval valued fuzzy graphs and their generalizations including neutrosophic graphs. Mathematics 7(1):72
Jan N, Zedam L, Mahmood T, Ullah K, Davvaz B, Ali Z (2019b) An improved clustering algorithm for picture fuzzy graphs and its applications in human decision- making. Int J Fuzzy Syst. https://doi.org/10.1007/s40815-019-00634-w
Jan N, Ali Z, Ullah K, Mahmood T (2019c) Some generalized distance and similarity measures for picture hesitant fuzzy sets and their applications in building material recognition and multi-attribute decision making. Punjab Univ J Math 51(7):51–70
Karthick P, Narayanamoorthy S (2016) The intuitionistic fuzzy line graph model to investigate radio coverage network. Int J Pure Appl Math 109(10):79–87
Karunambigai MG, Akram M, Sivasankar S, Palanivel K (2017) Clustering algorithm for intuitionistic fuzzy graphs. Int J Unc Fuzzy Knowl Based Syst 25(3):367–383
Kaufmann A (1973) Introduction a la Theorie des sons-ensembles flous, 1. Applications à la classification et à la reconnaisance des formes, aux automates et aux systèmes, aux choix des critères, Masson Paris, pp 41–189. https://cds.cern.ch/record/104322
Keller AA (2007) Graph theory and economic models: from small to large size applications. Electron Notes Discrete Math 28:469–476
Kiss A (1991) An application of fuzzy graphs in database theory, Automata languages and programming systems (Salgotarjan 1990). Pure Math Appl Ser A 1:337–342
Kóczy LT (1992) Fuzzy graphs in the evaluation and optimization of networks. Fuzzy Sets Syst 46:307–319
Liu WJ (1990) On some systems of simultaneous equations in a completely distributive lattice. Inf Sci 50:185–196
Mahmood T, Ullah K, Khan Q, Jan N (2018) An approach towards decision making and medical diagnosis problems using the concept of spherical fuzzy SETS. Neural Comput Appl. https://doi.org/10.1007/s00521-018-3521-2
Matula DW (1972) k-components, clusters, and slicings in graphs. SIAM J Appl Math 22:459–480
Mordeson JN, Peng C-S (1993) Fuzzy intersection equations. Fuzzy Sets Syst 60:77–78
Mordeson JN, Nair PS (2000) Applications of fuzzy graphs. In: Mordeson JN, Nair PS (eds) Fuzzy graphs and fuzzy hypergraphs. Studies in fuzziness and soft computing, 46, Physica, Heidelberg
Mukherjee S (2012) Dijkstra’s algorithm for solving the shortest path problem on networks under intuitionistic fuzzy environment. J Math Model Algorithms 11(4):345–359
Myna R (2015) Application of fuzzy graph in traffic. Int J Sci Eng Res 6(2):1692–1696
Neumann T (2016) Routing planning as an application of graph theory with fuzzy logic. Trans Nav Int J Mar Navig Saf Sea Transp. https://doi.org/10.12716/1001.10.04.17
Parvathi R, Karunambigai MG (2006) Intuitionistic fuzzy graphs. In: Reusch B (ed) Computational intelligence, theory and applications. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-34783-6_15
Parvathi R, Karunambigai MG, Atanassov KT (2009a) Operations on intuitionistic fuzzy graphs. In: Proceedings of the IEEE international conference on fuzzy systems, IEEE, pp 1396–1401
Parvathi R, Thilagavathi S, Karunambigai MG (2009b) Intuitionistic fuzzy hypergraphs. Cybern Inf Technol 9(2):46–53
Rosenfeld A (1975) Fuzzy graphs. In: Zadeh LA, Fu KS, Shimura M (eds) Fuzzy sets and their applications. Academic Press, Cambridge, pp 77–95
Ross IC, Harary F (1959) A description of strengthening and weakening members of a group. Sociometry 22:139–147
Shirinivas SG, Vetrivel S, Elango NM (2010) Applications of graph theory in computer science an overview. Int J Eng Sci Technol 2(9):4610–4621
Singh P (2015) Correlation coefficients for picture fuzzy sets. J Intell Fuzzy Syst 28(2):591–604
Son LH (2017) Measuring analogousness in picture fuzzy sets: from picture distance measures to picture association measures. Fuzzy Optim Dec Mak 16:359–378
Takeda E, Nishida T (1976) An application of fuzzy graph to the problem concerning group structure. J Oper Res Soc Jpn 19:217–227
Thong PH (2016) Picture fuzzy clustering: a new computational intelligence method. Soft Comput 20(9):3549–3562
Ullah K, Ali Z, Jan N, Mahmood T, Maqsood S (2018a) Picture hesitant fuzzy set and its applications. Tech J 23(04):84–95
Ullah K, Mahmood T, Jan N (2018b) Similarity measures for t-spherical fuzzy sets with applications in pattern recognition. Symmetry 10(6):193
Ullah K, Mahmood T, Jan N, Ali Z (2018c) A note on geometric aggregation operators in spherical fuzzy environment and its application in multi-attribute decision making. J Eng Appl Sci. https://doi.org/10.25211/jeas.v37i2.2871
Ullah K, Hassan N, Mahmood T, Jan N, Hassan M (2019a) evaluation of investment policy based on multi-attribute decision-making using interval valued T-spherical fuzzy aggregation operators. Symmetry. https://doi.org/10.3390/sym11030357
Ullah K, Garg H, Mahmood T, Jan N, Ali Z (2019b) Correlation coefficients for T-spherical fuzzy sets and their applications in clustering and multi-attribute decision making. Soft Comput. https://doi.org/10.1007/s00500-019-03993-6
Wei G (2017) Picture fuzzy aggregation operators and their application to multiple attribute decision making. J Intell Fuzzy Syst 33(2):713–724
Xu J (1997) The use of fuzzy graphs in chemical structure research. In: Rouvry DH (ed) Fuzzy logic in chemistry. Academic Press, Cambridge, pp 249–282
Xu ZS, Chen J (2008) An overview of distance and similarity measures of intuitionistic fuzzy sets. Int J Uncertain Fuzz Knowl-Based Syst 16(04):529–555
Yeh RT, Bang SY (1975) Fuzzy relations, fuzzy graphs, and their applications to clustering analysis. In: Zadeh LA, Fu KS, Shimura M (eds) Fuzzy sets and their applications. Academic Press, Cambridge, pp 125–149
Zadeh AL (1965) Fuzzy sets. Inf Control 8:338–353
Zuo C, Pal A, Dey A (2019) New concepts of picture fuzzy graphs with application. Mathematics 7(5):470. https://doi.org/10.3390/math7050470
Author information
Authors and Affiliations
Corresponding authors
Ethics declarations
Conflict of interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Additional information
Communicated by V. Loia.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Koczy, L.T., Jan, N., Mahmood, T. et al. Analysis of social networks and Wi-Fi networks by using the concept of picture fuzzy graphs. Soft Comput 24, 16551–16563 (2020). https://doi.org/10.1007/s00500-020-04959-9
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00500-020-04959-9