Selective Categories and Linear Canonical Relations

A construction of Wehrheim and Woodward circumvents the problem that compositions of smooth canonical relations are not always smooth, building a category suitable for functorial quantization. To apply their construction to more examples, we introduce a notion of highly selective category, in which only certain morphisms and certain pairs of these morphisms are "good". We then apply this notion to the category $\mathbf{SLREL}$ of linear canonical relations and the result ${\rm WW}(\mathbf{SLREL})$ of our version of the WW construction, identifying the morphisms in the latter with pairs $(L,k)$ consisting of a linear canonical relation and a nonnegative integer. We put a topology on this category of indexed linear canonical relations for which composition is continuous, unlike the composition in $\mathbf{SLREL}$ itself. Subsequent papers will consider this category from the viewpoint of derived geometry and will concern quantum counterparts

Visualization of Metabolic Networks

The metabolism constitutes the universe of biochemical reactions taking place in a cell of an organism. These processes include the synthesis, transformation, and degradation of molecules for an organism to grow, to reproduce and to interact with its environment. A good way to capture the complexity of these processes is the representation as metabolic network, in which sets of molecules are transformed into products by a chemical reaction, and the products are being processed further. The underlying graph model allows a structural analysis of this network using established graphtheoretical algorithms on the one hand, and a visual representation by applying layout algorithms combined with information visualization techniques on the other. In this thesis we will take a look at three different aspects of graph visualization within the context of biochemical systems: the representation and interactive exploration of static networks, the visual analysis of dynamic networks, and the comparison of two network graphs. We will demonstrate, how established infovis techniques can be combined with new algorithms and applied to specific problems in the area of metabolic network visualization. We reconstruct the metabolic network covering the complete set of chemical reactions present in a generalized eucaryotic cell from real world data available from a popular metabolic pathway data base and present a suitable data structure. As the constructed network is very large, it is not feasible for the display as a whole. Instead, we introduce a technique to analyse this static network in a top-down approach starting with an overview and displaying detailed reaction networks on demand. This exploration method is also applied to compare metabolic networks in different species and from different resources. As for the analysis of dynamic networks, we present a framework to capture changes in the connectivity as well as changes in the attributes associated with the network’s elements

Explorative Graph Visualization

Netzwerkstrukturen (Graphen) sind heutzutage weit verbreitet. Ihre Untersuchung dient dazu, ein besseres Verständnis ihrer Struktur und der durch sie modellierten realen Aspekte zu gewinnen. Die Exploration solcher Netzwerke wird zumeist mit Visualisierungstechniken unterstützt. Ziel dieser Arbeit ist es, einen Überblick über die Probleme dieser Visualisierungen zu geben und konkrete Lösungsansätze aufzuzeigen. Dabei werden neue Visualisierungstechniken eingeführt, um den Nutzen der geführten Diskussion für die explorative Graphvisualisierung am konkreten Beispiel zu belegen.Network structures (graphs) have become a natural part of everyday life and their analysis helps to gain an understanding of their inherent structure and the real-world aspects thereby expressed. The exploration of graphs is largely supported and driven by visual means. The aim of this thesis is to give a comprehensive view on the problems associated with these visual means and to detail concrete solution approaches for them. Concrete visualization techniques are introduced to underline the value of this comprehensive discussion for supporting explorative graph visualization

Streaming, Local, and Multi­Level (Hyper)Graph Decomposition

(Hyper)Graph decomposition is a family of problems that aim to break down large (hyper)graphs into smaller sub(hyper)graphs for easier analysis. The importance of this lies in its ability to enable efficient computation on large and complex (hyper)graphs, such as social networks, chemical compounds, and computer networks. This dissertation explores several types of (hyper)graph decomposition problems, including graph partitioning, hypergraph partitioning, local graph clustering, process mapping, and signed graph clustering. Our main focus is on streaming algorithms, local algorithms and multilevel algorithms. In terms of streaming algorithms, we make contributions with highly efficient and effective algorithms for (hyper)graph partitioning and process mapping. In terms of local algorithms, we propose sub-linear algorithms which are effective in detecting high-quality local communities around a given seed node in a graph based on the distribution of a given motif. In terms of multilevel algorithms, we engineer high-quality multilevel algorithms for process mapping and signed graph clustering. We provide a thorough discussion of each algorithm along with experimental results demonstrating their superiority over existing state-of-the-art techniques. The results show that the proposed algorithms achieve improved performance and better solutions in various metrics, making them highly promising for practical applications. Overall, this dissertation showcases the effectiveness of advanced combinatorial algorithmic techniques in solving challenging (hyper)graph decomposition problems

A control for graph representation and interaction

Estágio realizado na ParadigmaXis e orientado pelo Eng.º Filipe CorreiaTese de mestrado integrado. Engenharia Informátca e Computação. Faculdade de Engenharia. Universidade do Porto. 200