Schematics of Graphs and Hypergraphs

Graphenzeichnen als ein Teilgebiet der Informatik befasst sich mit dem Ziel Graphen oder deren Verallgemeinerung Hypergraphen geometrisch zu realisieren. Beschränkt man sich dabei auf visuelles Hervorheben von wesentlichen Informationen in Zeichenmodellen, spricht man von Schemata. Hauptinstrumente sind Konstruktionsalgorithmen und Charakterisierungen von Graphenklassen, die für die Konstruktion geeignet sind. In dieser Arbeit werden Schemata für Graphen und Hypergraphen formalisiert und mit den genannten Instrumenten untersucht. In der Dissertation wird zunächst das „partial edge drawing“ (kurz: PED) Modell für Graphen (bezüglich gradliniger Zeichnung) untersucht. Dabei wird um Kreuzungen im Zentrum der Kante visuell zu eliminieren jede Kante durch ein kreuzungsfreies Teilstück (= Stummel) am Start- und am Zielknoten ersetzt. Als Standard hat sich eine PED-Variante etabliert, in der das Längenverhältnis zwischen Stummel und Kante genau 1⁄4 ist (kurz: 1⁄4-SHPED). Für 1⁄4-SHPEDs werden Konstruktionsalgorithmen, Klassifizierung, Implementierung und Evaluation präsentiert. Außerdem werden PED-Varianten mit festen Knotenpositionen und auf Basis orthogonaler Zeichnungen erforscht. Danach wird das BUS Modell für Hypergraphen untersucht, in welchem Hyperkanten durch fette horizontale oder vertikale – als BUS bezeichnete – Segmente repräsentiert werden. Dazu wird eine vollständige Charakterisierung von planaren Inzidenzgraphen von Hypergraphen angegeben, die eine planare Zeichnung im BUS Modell besitzen, und diverse planare BUS-Varianten mit festen Knotenpositionen werden diskutiert. Zum Schluss wird erstmals eine Punktmenge von subquadratischer Größe angegeben, die eine planare Einbettung (Knoten werden auf Punkte abgebildet) von 2-außenplanaren Graphen ermöglicht

Structural sparsity of complex networks: bounded expansion in random models and real-world graphs

This research establishes that many real-world networks exhibit bounded expansion, a strong notion of structural sparsity, and demonstrates that it can be leveraged to design efficient algorithms for network analysis. Specifically, we give a new linear-time fpt algorithm for motif counting and linear time algorithms to compute localized variants of several centrality measures. To establish structural sparsity in real-world networks, we analyze several common network models regarding their structural sparsity. We show that, with high probability, (1) graphs sampled with a prescribed sparse degree sequence; (2) perturbed bounded-degree graphs; (3) stochastic block models with small probabilities; result in graphs of bounded expansion. In contrast, we show that the Kleinberg and the Barabási–Albert model have unbounded expansion. We support our findings with empirical measurements on a corpus of real-world networks

A Distance-preserving Matrix Sketch

Visualizing very large matrices involves many formidable problems. Various popular solutions to these problems involve sampling, clustering, projection, or feature selection to reduce the size and complexity of the original task. An important aspect of these methods is how to preserve relative distances between points in the higher-dimensional space after reducing rows and columns to fit in a lower dimensional space. This aspect is important because conclusions based on faulty visual reasoning can be harmful. Judging dissimilar points as similar or similar points as dissimilar on the basis of a visualization can lead to false conclusions. To ameliorate this bias and to make visualizations of very large datasets feasible, we introduce two new algorithms that respectively select a subset of rows and columns of a rectangular matrix. This selection is designed to preserve relative distances as closely as possible. We compare our matrix sketch to more traditional alternatives on a variety of artificial and real datasets.Comment: 38 pages, 13 figure

Tree Topology Estimation

<p>Tree-like structures are fundamental in nature. A wide variety of two-dimensional imaging techniques allow us to image trees. However, an image of a tree typically includes spurious branch crossings and the original relationships of ancestry among edges may be lost. We present a methodology for estimating the most likely topology of a rooted, directed, three-dimensional tree given a single two-dimensional image of it. We regularize this inverse problem via a prior parametric tree-growth model that realistically captures the morphology of a wide variety of trees. We show that the problem of estimating the optimal tree has linear complexity if ancestry is known, but is NP-hard if it is lost. For the latter case, we present both a greedy approximation algorithm and a heuristic search algorithm that effectively explore the space of possible trees. Experimental results on retinal vessel, plant root, and synthetic tree datasets show that our methodology is both accurate and efficient.</p>Dissertatio

Impedance Transformers

Non

3D to 2D surface mesh parameterisation for the purposes of unstructured transmission line modelling method simulations

Small scale fabrication processes have led to the advent of very thin flexible devices such as RFID tags, flexible PCBs and smart clothing. In a geometrical sense, these present themselves as curved two dimensional surfaces embedded in a three dimensional domain. When simulating electromagnetic behaviour on these surfaces at low frequencies, a full 3D field model is not always necessary. Using 3D algorithms to solve these problems can result in a large portion of the computer memory and runtime being used to mesh and simulate areas of the domain that present little electromagnetic activity. The theme of this thesis is concerned with the improvement of the runtime and memory consumption of electromagnetic simulations of these surfaces. The main contributions of this work are presented as an investigation into the feasibility of applying a 2D Unstructured Transmission Line Modelling method (UTLM) simulation to open, curved surfaces embedded in 3D space, by providing a one-to-one mapping of the geometry to a 2D flat plane. First, an investigation into the various methods of how a computer represents unstructured meshes in its memory is presented, and how this affects the runtime of the simulation. The underlying mesh data structures used to represent the geometrical problem space can have a huge impact on the efficiency and memory consumption of the simulation. This investigation served to demonstrate that it is not just simply the optimisation of the simulation algorithms that facilitate improvements to the runtime and memory consumption of a simulation. How a computer understands the connectivity of the mesh can have far greater impacts to the computational resources available. The concepts of surface parameterisation are then introduced; a process of mapping curved surfaces embedded in a three dimensional domain to a flat two dimensional plane. By providing a one-to-one mapping of the geometry from the 3D domain to the 2D flat plane, a low frequency 2D unstructured TLM simulation can be applied, negating the need for 3D algorithms. Because this mapping is one-to-one, the results of the simulation can then be mapped back to 3D space for visualisation. Parameterisations will almost always introduce distortion to angle and area, and minimising this distortion is paramount to maintaining an accurate simulation. Test cases were used to measure the extent of this distortion, and the investigation concluded that Angle Based Flattening (ABF) and Least Squares Conformal Mapping (LSCM) methods resulted in the best quality parameterisations. Simulations were then conducted on these test cases as a demonstration of how UTLM can be performed on 2D surfaces, embedded in a 3D domain

3D to 2D surface mesh parameterisation for the purposes of unstructured transmission line modelling method simulations

Small scale fabrication processes have led to the advent of very thin flexible devices such as RFID tags, flexible PCBs and smart clothing. In a geometrical sense, these present themselves as curved two dimensional surfaces embedded in a three dimensional domain. When simulating electromagnetic behaviour on these surfaces at low frequencies, a full 3D field model is not always necessary. Using 3D algorithms to solve these problems can result in a large portion of the computer memory and runtime being used to mesh and simulate areas of the domain that present little electromagnetic activity. The theme of this thesis is concerned with the improvement of the runtime and memory consumption of electromagnetic simulations of these surfaces. The main contributions of this work are presented as an investigation into the feasibility of applying a 2D Unstructured Transmission Line Modelling method (UTLM) simulation to open, curved surfaces embedded in 3D space, by providing a one-to-one mapping of the geometry to a 2D flat plane. First, an investigation into the various methods of how a computer represents unstructured meshes in its memory is presented, and how this affects the runtime of the simulation. The underlying mesh data structures used to represent the geometrical problem space can have a huge impact on the efficiency and memory consumption of the simulation. This investigation served to demonstrate that it is not just simply the optimisation of the simulation algorithms that facilitate improvements to the runtime and memory consumption of a simulation. How a computer understands the connectivity of the mesh can have far greater impacts to the computational resources available. The concepts of surface parameterisation are then introduced; a process of mapping curved surfaces embedded in a three dimensional domain to a flat two dimensional plane. By providing a one-to-one mapping of the geometry from the 3D domain to the 2D flat plane, a low frequency 2D unstructured TLM simulation can be applied, negating the need for 3D algorithms. Because this mapping is one-to-one, the results of the simulation can then be mapped back to 3D space for visualisation. Parameterisations will almost always introduce distortion to angle and area, and minimising this distortion is paramount to maintaining an accurate simulation. Test cases were used to measure the extent of this distortion, and the investigation concluded that Angle Based Flattening (ABF) and Least Squares Conformal Mapping (LSCM) methods resulted in the best quality parameterisations. Simulations were then conducted on these test cases as a demonstration of how UTLM can be performed on 2D surfaces, embedded in a 3D domain

The modular structure of brain functional connectivity networks: a graph theoretical approach

Complex networks theory offers a framework for the analysis of brain functional connectivity as measured by magnetic resonance imaging. Within this approach the brain is represented as a graph comprising nodes connected by links, with nodes corresponding to brain regions and the links to measures of inter-regional interaction. A number of graph theoretical methods have been proposed to analyze the modular structure of these networks. The most widely used metric is Newman's Modularity, which identifies modules within which links are more abundant than expected on the basis of a random network. However, Modularity is limited in its ability to detect relatively small communities, a problem known as ``resolution limit''. As a consequence, unambiguously identifiable modules, like complete sub-graphs, may be unduly merged into larger communities when they are too small compared to the size of the network. This limit, first demonstrated for Newman's Modularity, is quite general and affects, to a different extent, all methods that seek to identify the community structure of a network through the optimization of a global quality function. Hence, the resolution limit may represent a critical shortcoming for the study of brain networks, and is likely to have affected many of the studies reported in the literature. This work pioneers the use of Surprise and Asymptotical Surprise, two quality functions rooted in probability theory that aims at overcoming the resolution limit for both binary and weighted networks. Hereby, heuristics for their optimization are developed and tested, showing that the resulting optimal partitioning can highlight anatomically and functionally plausible modules from brain connectivity datasets, on binary and weighted networks. This novel approach is applied to the partitioning of two different human brain networks that have been extensively characterized in the literature, to address the resolution-limit issue in the study of the brain modular structure. Surprise maximization in human resting state networks revealed the presence of a rich structure of modules with heterogeneous size distribution undetectable by current methods. Moreover, Surprise led to different, more accurate classification of the network's connector hubs, the elements that integrate the brain modules into a cohesive structure. In synthetic networks, Asymptotical Surprise showed high sensitivity and specificity in the detection of ground-truth structures, particularly in the presence of noise and variability such as those observed in experimental functional MRI data. Finally, the methodological advances hereby introduced are shown to be a helpful tool to better discern differences between the modular organization of functional connectivity of healthy subjects and schizophrenic patients. Importantly, these differences may point to new clinical hypotheses on the etiology of schizophrenia, and they would have gone unnoticed with resolution-limited methods. This may call for a revisitation of some of the current models of the modular organization of the healthy and diseased brain