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Pretopology, a mathematical tool for structuring complex systems : methods, algorithms and applications

Abstract : When we study a complex system it is natural to be interested in the structure of its elements, as opposed to just caring about the individual characteristics of the components. The traditional way to deal with this issue has been through a characterization of the structure by the use of graphs, where nodes represent the components of the system, and an edge exists between two of them if there is a relation connecting them. The theory is nevertheless only appropriate for the description of systems with binary underlying relations between the components. We propose here the use of pretopology to effectively treat the case where connections exist between an element and a group. Our contributions include a formalization of a pretopological space in terms of a simple group of rules over a set of networks. This allow us to economically store a pretopological space, and to effectively study the complexity of the algorithms that have been proposed. Some of these algorithms were improved. Some applications are developed. The first of them is a Python library where all of the previously reviewed algorithms are implemented. After presenting in a general manner the faculties of the library, we use it to study some diffusion phenomena with the help of some agent based models. We finish using some of the notions introduced here to develop a clusterization algorithm, that not only performs on a par with the state of the art on some artificial geometrical data, but also has the advantage of being immediately generalizable to non-metrical spaces.
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Submitted on : Wednesday, May 27, 2020 - 10:00:06 PM
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  • HAL Id : tel-02637029, version 1

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Julio Laborde. Pretopology, a mathematical tool for structuring complex systems : methods, algorithms and applications. General Topology [math.GN]. Université Paris sciences et lettres, 2019. English. ⟨NNT : 2019PSLEP010⟩. ⟨tel-02637029⟩

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