Skip to main content
SpringerLink
Log in

Hierarchical Decomposition by Means of Fuzzy Simplicial Complexes

  • Conference paper
  • First Online:
Intelligent and Fuzzy Techniques for Emerging Conditions and Digital Transformation (INFUS 2021)

Part of the book series: Lecture Notes in Networks and Systems ((LNNS,volume 307))

Included in the following conference series:

  • 1027 Accesses

Abstract

The approach emerging from the algebraic topology known by the Q-analysis (or Polyhedral Dynamics) method proposed by R.H. Atkin for the analysis of the structures of complex systems as well as for the measurement of the structural complexity of systems, has given rise to several works and applications in various fields. In this paper, we specifically develop a new hierarchical analysis method in the field of topological data analysis and explore its promising implications and results for the study of complex systems. Our method concerns systems governed by fuzzy connectivities and generalizes in some way the classical polyhedral dynamics approach and it is based on fuzzy path algebra and simplicial complexes. The presented system is a general framework that can be applied to various case studies of complex systems to aid in understanding their topology and logic. Results present, in a general way, the strengths and contribution of this method as well as its relevance for the analysis of complex systems.

This work was partially funded by Ministry of Equipment, Transport, Logistics and Water – Kingdom of Morocco, The National Road Safety Agency (NARSA) and National Center for Scientific and Technical Research (CNRST). Road Safety Research Program# An intelligent reactive abductive system and intui-tionist fuzzy logical reasoning for dangerousness of driver-pedestrians interactions analysis: Development of new pedestrians’ exposure to risk of road accident measures.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 169.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 219.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

References

  1. Alexandroff, P.S.: Elementary Concepts of Topology. Dover Publications, N.Y. (1961)

    MATH  Google Scholar 

  2. Atkin, R.H.: An algebra for patterns on a complex, II. Int. J. Man Mach. Stud. 8, 483–498 (1976)

    Article  MathSciNet  Google Scholar 

  3. Atkin, R.H.: Combinatorial Connectivities in Social Systems. Birkhauser, Basel (1977)

    Book  Google Scholar 

  4. Atkin, R.H.: From cohomology in physics to q-connectivity in social science. Int. J. Man Mach. Stud. 4, 139–167 (1972)

    Article  MathSciNet  Google Scholar 

  5. Atkin, R.H.: Mathematical Structure in Human Affairs. Heinemann, London (1974)

    Google Scholar 

  6. Bloch, I.: Fuzzy connectivity and mathematical morphology. Pattern Recogn. Lett. 14, 483–488 (1993). (North-Holland)

    Google Scholar 

  7. Boulmakoul, A., Besri, Z.: Performing enterprise organizational structure redesign through structural analysis and simplicial complexes framework. Open Oper. Res. J. 7, 11–24 (2013)

    MathSciNet  MATH  Google Scholar 

  8. Boulmakoul, A., Idri, A., Marghoubi, R.: Closed frequent itemsets mining and struc-turing association rules based on q-analysis. In: 2007 IEEE International Symposiumon Signal Processing and Information Technology, pp. 519–524. IEEE (2007)

    Google Scholar 

  9. Chamorro-Martínez, J., Sánchez, D., Soto-Hidalgo, J.M., Martínez-Jiménez, P.M.: A discussion on fuzzy cardinality and quantification. Some Appl. Image Process. Fuzzy Sets Syst. 257, 85–101 (2014)

    Article  Google Scholar 

  10. Chazal, F., Michel, B.: An introduction to topological data analysis: fundamental and practical aspects for data scientists. arXiv preprint (2017)

    Google Scholar 

  11. Horak, D., Maleti ́c, S., Rajkovi ́c, M.: Persistent homology of complex networks. J. Stat. Mech.: Theory Exper. 2009(03), P03034 (2009)

    Google Scholar 

  12. Edelsbrunner, H., Letscher, D., Zomorodian, A.: Topological persistence and simplification. In: Proceedings 41st Annual Symposium on Foundations of Computer Science, pp. 454–463. IEEE (2000)

    Google Scholar 

  13. El-Ghoul, M., El-Zohny, H., Radwan, S.: Fuzzy incidence matrix of fuzzy simplicial complexes and its folding. Chaos, Solitons Fractals 13(9), 1827–1833 (2002)

    Article  MathSciNet  Google Scholar 

  14. El-Ghoul, M.: Folding of fuzzy torus and fuzzy graphs Fuzzy Sets and System. Germany 80, 389–396 (1996)

    MATH  Google Scholar 

  15. Faridi, S.: The facet ideal of a simplicial complex. Manuscripta Math. 109(2), 159–174 (2002)

    Article  MathSciNet  Google Scholar 

  16. Ferrario, D.L., Piccinini, R.A.: Simplicial Structures in Topology, CMS Books in Mathematics Springer (2011). https://doi.org/10.1007/978-1-4419-7236-1

  17. Giblin, P.J.: Graphs, Surfaces, and Homology: An Introduction to Algebraic Topology. Wiley, New York (1977)

    MATH  Google Scholar 

  18. Glazek, K.: A Guide to the Literature on Semirings and their Applications in Mathematics and Information Sciences: With Complete Bibliography, Springer Science & Business Media, June 29 (2013). https://doi.org/10.1007/978-94-015-9964-1

  19. Jonsson, J.: Simplicial Complexes of Graphs, Lecture Notes in Mathematics. Springer (2008). https://doi.org/10.1007/978-3-540-75859-4

  20. Maguerra, S., Boulmakoul, A., Karim, L., Badir, H.: A distributed execution pipeline for clustering trajectories based on a fuzzy similarity relation. Algorithms 12(2), 29 (2019)

    Article  Google Scholar 

  21. Maleti ́c, S., Zhao, Y., Rajkovi ́c, M.: Persistent topological features of dynamical systems. Chaos: Interdiscip. J. Nonlinear Sci. 26(5), 053105 (2016)

    Google Scholar 

  22. Mordeson, J.N., Nair, P.S. (eds.): Fuzzy Graphs and Fuzzy Hypergraphs. Physica-Verlag HD, Heidelberg (2000)

    MATH  Google Scholar 

  23. Otter, N., Porter, M.A., Tillmann, U., Grindrod, P., Harrington, H.A.: A roadmap for the computation of persistent homology. EPJ Data Science 6(1), 1–38 (2017). https://doi.org/10.1140/epjds/s13688-017-0109-5

    Article  Google Scholar 

  24. Slobodan, M., Milan, R.: Combinatorial Laplacian and entropy of simplicial complexes associated with complex networks. Euro. Phys. J. ST 212, 77–97 (2012)

    Article  Google Scholar 

  25. Topaz, C.M., Ziegelmeier, L., Halverson, T.: Topological data analysis of biological aggregation models. PLoS ONE 10(5), e0126383 (2015)

    Google Scholar 

  26. Lin, T.Y., Chiang, I.J.: A simplicial complex, a hypergraph, structure in the latent semantic space of document clustering. Int. J. Approx. Reason. 40(1–2), 55–80 (2005)

    Article  MathSciNet  Google Scholar 

  27. Vick, J.W.: Homology Theory: An Introduction to Algebraic Topology (Graduate Texts in Mathematics) (v. 145) 2nd edn. Springer (1994). https://doi.org/10.1007/978-1-4612-0881-5

  28. Wasserman, L.: Topological data analysis. Ann. Rev. Stat. Appl. 5, 501–532 (2018)

    Google Scholar 

  29. Zadeh, L.A.: Computational approach to fuzzy quantifiers in natural languages. Comput. Math. Appl. 9(1), 149–184 (1983)

    Google Scholar 

  30. Zadeh, L.A.: Fuzzy sets, Information and Control, vol. 8, Issue 3, June 1965, pp. 338–353 (1965)

    Google Scholar 

  31. Zadeh, L.A.: Similarity relations and fuzzy orderings. Inf. Sci. 3(2), 177–200 (1971)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, FSTM, Hassan II University of Casablanca, Casablanca, Morocco

    Azedine Boulmakoul, Hafsaa Ouifak & Lamia Karim

  2. LISA Laboratory ENSA Berrechid, Hassan 1st University, Casablanca, Morocco

    Lamia Karim

  3. LIG/ MRIM, CNRS, University Grenoble Alpes, Grenoble, France

    Azedine Boulmakoul & Ahmed Lbath

Authors
  1. Azedine Boulmakoul
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hafsaa Ouifak
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Lamia Karim
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Ahmed Lbath
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Industrial Engineering, Istanbul Technical University, Istanbul, Turkey

    Cengiz Kahraman

  2. Industrial Engineering Department, Yildiz Technical University, Istanbul, Turkey

    Selcuk Cebi

  3. Department of Industrial Engineering, Istanbul Technical University, Istanbul, Turkey

    Sezi Cevik Onar

  4. Department of Industrial Engineering, Istanbul Technical University, Istanbul, Turkey

    Basar Oztaysi

  5. Industrial Engineering Department, Galatasaray University, Istanbul, Turkey

    A. Cagri Tolga

  6. Department of Industrial Engineering, Istanbul Technical University, Istanbul, Turkey

    Irem Ucal Sari

Rights and permissions

Reprints and permissions

Copyright information

© 2022 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Boulmakoul, A., Ouifak, H., Karim, L., Lbath, A. (2022). Hierarchical Decomposition by Means of Fuzzy Simplicial Complexes. In: Kahraman, C., Cebi, S., Cevik Onar, S., Oztaysi, B., Tolga, A.C., Sari, I.U. (eds) Intelligent and Fuzzy Techniques for Emerging Conditions and Digital Transformation. INFUS 2021. Lecture Notes in Networks and Systems, vol 307. Springer, Cham. https://doi.org/10.1007/978-3-030-85626-7_26

Download citation

Publish with us

Policies and ethics

Search

Navigation