- Titel
- Continuous methods for discrete structures
- Zitierfähige Url:
- https://nbn-resolving.org/urn:nbn:de:bsz:15-qucosa2-980993
- Datum der Einreichung
- 30.09.2024
- Datum der Verteidigung
- 25.02.2025
- Abstract (EN)
- We develop rigorous mathematical tools to explore the structure and underlying geometry of networks, drawing on ideas at the intersection of the continuous and discrete worlds. Our contributions encompass spectral graph theory, discrete Ricci curvatures, and graph and hypergraph limit theory. First, we advance the understanding of the spectrum and eigenfunctions of the non-backtracking Laplacian, a recently introduced Laplace operator for graphs that is related to non-backtracking random walks. Second, we address the problem of constructing graphs with a specified discrete Ricci curvature and explore the space of such graphs. In particular, we derive a finite set of rewiring moves that connects the space of all graphs with a given Forman-Ricci curvature and degree sequence. Finally, as the main mathematical contribution, we extend graph limit theory by developing new notions of convergence and limits for sequences of graphs and generalizations. Specifically, we study two concepts of edge-decorated graph limits, probability graphons and P-variables, showing several properties of these objects and proving the equivalence of these two frameworks. Additionally, we develop limit theories for hypergraphs, higher-order generalizations of graphs, and analyze their properties.
- Andere Ausgabe
- There is no going back: properties of the non-backtracking Laplacian
DOI: https://doi.org/10.1016/j.laa.2023.10.014
Link: https://www.sciencedirect.com/science/article/pii/S0024379523003889 - Exploring the space of graphs with fixed discrete curvatures
DOI: 10.1088/2632-072X/ad679f
Link: https://iopscience.iop.org/article/10.1088/2632-072X/ad679f/meta - A measure-theoretic representation of graphs
Link: https://link.springer.com/article/10.1007/s10998-023-00536-3
DOI: https://doi.org/10.1007/s10998-023-00536-3 - Probability graphons: the right convergence point of view
Link: https://arxiv.org/abs/2407.05998 - Probability graphons and P-variables: two equivalent viewpoints for dense weighted graph limits
Link: https://arxiv.org/abs/2408.07572 - Action convergence of general hypergraphs and tensors
Link: https://arxiv.org/abs/2308.00226 - Freie Schlagwörter (EN)
- Spectral graph theory, Discrete curvature, Graph limits, Hypergraphs
- Klassifikation (DDC)
- 500
- Den akademischen Grad verleihende / prüfende Institution
- Universität Leipzig, Leipzig
- Version / Begutachtungsstatus
- publizierte Version / Verlagsversion
- URN Qucosa
- urn:nbn:de:bsz:15-qucosa2-980993
- Veröffentlichungsdatum Qucosa
- 29.07.2025
- Dokumenttyp
- Dissertation
- Sprache des Dokumentes
- Englisch
- Lizenz / Rechtehinweis
CC BY 4.0