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The enumeration of paths in networks is a central topic in enumerative combinatorics. In particular, the study of paths taking steps in a finite admissible set and constrained to a certain region has been the subject of sustained effort over the last twenty years, extending the historical example of Dyck paths. In particular, the systematic use of catalytic variable equations to address this problem allows us to apply numerous techniques from enumerative combinatorics, formal computation, probability theory, analysis, and Galois theory to differences, in order to classify the generating series of such paths according to an algebraic-differential hierarchy. This collaboration led to a remarkable initial classification of the generating series of paths from small-step models, initiated by the work of Fayolle, Iasnogorodski, and Malyshev, and later by Bousquet-Mélou and Mishna, and completed by many other authors from the above-mentioned disciplines. Since then, numerous efforts have been made to extend the tools of this initial classification in order to deal with more complex cases. It is in this context that the thesis is situated, built around two different extensions, the first concerning large-step models, the second concerning paths with interactive edges. 

Amphi LaBRI