Path homology is a homology theory on directed graphs introduced by Grigor’yan, Lin, Muranov, and Yau in the early 2010s, aiming to extract topological and algebraic invariants that are sensitive to the orientation of edges. Mémoli and Chowdhury then defined persistent path homology on weighted directed graphs, and showed its stability under a certain network distance. We will introduce these objects and show the existence of the persistence diagram for a continuous model of directed Rips weighted graphs.

This internship took place at the Laboratoire de Mathématiques d’Orsay.

References: 

[1] Grigorian, Alexander. (2022). Advances in path homology theory of digraphs. Notices of the International Consortium of Chinese Mathematicians. https://dx.doi.org/10.4310/ICCM.2022.v10.n2.a7

[2] Chowdhury, Samir & Mémoli, Facundo. (2018). Persistent Path Homology of Directed Networks. 10.1137/1.9781611975031.75.