- Titel
- Topological and Geometric Methods with a View Towards Data Analysis
- Zitierfähige Url:
- https://nbn-resolving.org/urn:nbn:de:bsz:15-qucosa2-788338
- Datum der Einreichung
- 13.08.2021
- Datum der Verteidigung
- 12.01.2022
- Abstract (EN)
- In geometry, various tools have been developed to explore the topology and other features of a manifold from its geometrical structure. Among the two most powerful ones are the analysis of the critical points of a function, or more generally, the closed orbits of a dynamical system defined on the manifold, and the evaluation of curvature inequalities. When any (nondegenerate) function has to have many critical points and with different indices, then the topology must be rich, and when certain curvature inequalities hold throughout the manifold, that constrains the topology. It has been observed that these principles also hold for metric spaces more general than Riemannian manifolds, and for instance also for graphs. This thesis represents a contribution to this program. We study the relation between the closed orbits of a dynamical system and the topology of a manifold or a simplicial complex via the approach of Floer. And we develop notions of Ricci curvature not only for graphs, but more generally for, possibly directed, hypergraphs, and we draw structural consequences from curvature inequalities. It includes methods that besides their theoretical importance can be used as powerful tools for data analysis. This thesis has two main parts; in the first part we have developed topological methods based on the dynamic of vector fields defined on smooth as well as discrete structures. In the second part, we concentrate on some curvature notions which already proved themselves as powerful measures for determining the local (and global) structures of smooth objects. Our main motivation here is to develop methods that are helpful for the analysis of complex networks. Many empirical networks incorporate higher-order relations between elements and therefore are naturally modeled as, possibly directed and/or weighted, hypergraphs, rather than merely as graphs. In order to develop a systematic tool for the statistical analysis of such hypergraphs, we propose a general definition of Ricci curvature on directed hypergraphs and explore the consequences of that definition. The definition generalizes Ollivier’s definition for graphs. It involves a carefully designed optimal transport problem between sets of vertices. We can then characterize various classes of hypergraphs by their curvature. In the last chapter, we show that our curvature notion is a powerful tool for determining complex local structures in a variety of real and random networks modeled as (directed) hypergraphs. Furthermore, it can nicely detect hyperloop structures; hyperloops are fundamental in some real networks such as chemical reactions as catalysts in such reactions are faithfully modeled as vertices of directed hyperloops. We see that the distribution of our curvature notion in real networks deviates from random models.
- Verweis
- Ollivier Ricci curvature of directed hypergraphs
Link: https://www.nature.com/articles/s41598-020-68619-6
DOI: https://doi.org/10.1038/s41598-020-68619-6 - Ricci curvature of random and empirical directed hypernetworks
Link: https://appliednetsci.springeropen.com/articles/10.1007/s41109-020-00309-8
DOI: https://doi.org/10.1007/s41109-020-00309-8 - Edge-based analysis of networks: curvatures of graphs and hypergraphs
Link: https://link.springer.com/article/10.1007/s12064-020-00328-0
DOI: https://doi.org/10.1007/s12064-020-00328-0 - Floer Homology: From Generalized Morse-Smale Dynamical Systems to Forman's Combinatorial Vector Fields
DOI: https://doi.org/10.48550/arXiv.2105.02567
Link: https://arxiv.org/pdf/2105.02567.pdf - Freie Schlagwörter (EN)
- Floer Homology, Ricci curvature, Data Analysis, complex networks, optimal transport
- Klassifikation (DDC)
- 500
- Den akademischen Grad verleihende / prüfende Institution
- Universität Leipzig, Leipzig
- Version / Begutachtungsstatus
- angenommene Version / Postprint / Autorenversion
- URN Qucosa
- urn:nbn:de:bsz:15-qucosa2-788338
- Veröffentlichungsdatum Qucosa
- 12.04.2022
- Dokumenttyp
- Dissertation
- Sprache des Dokumentes
- Englisch
- Lizenz / Rechtehinweis
CC BY 4.0