Hierarchical Decomposition by Means of Fuzzy Simplicial Complexes

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.