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.
| Identifer | oai:union.ndltd.org:DRESDEN/oai:qucosa:de:qucosa:78833 |
| Date | 12 April 2022 |
| Creators | Eidi, Marzieh |
| Contributors | Universität Leipzig |
| Source Sets | Hochschulschriftenserver (HSSS) der SLUB Dresden |
| Language | English |
| Detected Language | English |
| Type | info:eu-repo/semantics/acceptedVersion, doc-type:doctoralThesis, info:eu-repo/semantics/doctoralThesis, doc-type:Text |
| Rights | info:eu-repo/semantics/openAccess |
| Relation | https://doi.org/10.1038/s41598-020-68619-6, https://doi.org/10.1007/s41109-020-00309-8, https://doi.org/10.1007/s12064-020-00328-0, https://doi.org/10.48550/arXiv.2105.02567 |
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