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Simplicial complexes and complex systems. (English) Zbl 1420.55033

Summary: We provide a short introduction to the field of topological data analysis (TDA) and discuss its possible relevance for the study of complex systems. TDA provides a set of tools to characterise the shape of data, in terms of the presence of holes or cavities between the points. The methods, based on the notion of simplicial complexes, generalise standard network tools by naturally allowing for many-body interactions and providing results robust under continuous deformations of the data. We present strengths and weaknesses of current methods, as well as a range of empirical studies relevant to the field of complex systems, before identifying future methodological challenges to help understand the emergence of collective phenomena.

MSC:

55U10 Simplicial sets and complexes in algebraic topology
18G30 Simplicial sets; simplicial objects in a category (MSC2010)
54C56 Shape theory in general topology
62N02 Estimation in survival analysis and censored data
37F20 Combinatorics and topology in relation with holomorphic dynamical systems
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