Das Studium von Transportmechanismen in komplexen Organismen stellt eine zentrale Herausforderung dar, nicht nur in medizinischen und biologischen Disziplinen, sondern auch zunehmend in der Physik und Netzwerktheorie. Insbesondere sind bionisch inspirierte Designprinzipien zunehmend relevant, da sie zuverlässige Lösungsansätze zu verschiedenen theoretischen und technischen Problemen bieten. Herausstechend sind dabei vaskuläre Netzwerke in Säugetieren, deren Entwicklung auffällig stark auf Selbstorganisation beruhen und die korrekte Verteilung von Sauerstoff, Wasser, Blut oder Ähnlichem erlaubt. Dies wird erreicht durch ein komplexes biochemisches Signalsystem, welches an makroskopische Stimulationen, wie z. B. Reibung und Stress, gekoppelt ist. Die Morphogenese solcher Flussnetzwerke ist allerdings noch anderen Restriktionen unterworfen, da diese räumlich eingebettete Objekte darstellen. Sie sind als solche signifikant beschränkter in ihrer Skalierbarkeitund Dynamik. Diese Dissertation addressiert daher relevante Fragestellungen zur Charakterisierung von Netzwerken und der Morphogenesesimulationen von drei-dimensional eingebetteten Netzwerken Die Schlüsselmechanismen auf die wir uns hier konzentrieren sind Flussfluktuationen, Interaktionen zwischen Paarstrukturen und die Aufnahme von Nährstoffen. Zu Beginn zeigen wir, wie sich konventionelle Ansätze zu Flussfluktuationen als allgemeine Einparametermodelle darstellen lassen. Wir demonstrieren damit den kontinuierlichen Übergang zu zunehmend vernetzten Strukturen und indizieren Topologieabhängigkeiten der Plexus in Anbetracht dieses Übergangs. Darauf aufbauend formulieren wir ein neues Adaptationsmodell für ineinander verwobene Gefäßnetzwerke wie sie auch in der Leber, Bauchspeicheldrüse oder Niere vorkommen. Wir diskutieren anhand dieser Strukturen lokale Wechselwirkungen von dreidimensionalen Netzwerken. Dadurch können wir zeigen, dass repulsiv gekoppelte Netzwerke fluktuationsinduzierte Vernetzungen auflösen und attraktive Kopplungen einen neuen Mechanismus zur Erzeugung eben jener darstellen. Als nächstes verallgemeinern wir die Murray Regel für solch komplexe Wechselwirkungen und Fluktuationen. Die daraus abgeleiteten Relationen nutzen wir zur Regression der Modellparameter und testen diese an den Gefäßnetzwerken der Leber. Weiterhin verallgemeinern wir konventionelle Transportmodelle für die Nährstoffaufnahme in beliebigem Gewebe und testen diese in Morphogenesemodellen gegen die bekannten Ansätze zur Dissipationsminimierung. Hier zeigen sich komplexe Übergänge zwischen vernetzten Strukturen und unkonventionelles Phasenverhalten. Allerdings indizieren die Ergebnisse Widersprüche zu echten Kapillargefäßen und wir vermuten Adaptationsmethoden ohne Gefäßgrößenänderung als wahrscheinlicheren Mechanismus. Im Ausblick schlagen wir auf unseren Ergebnissen aufbauende Folgemodelle vor, welche die Modellierung komplexer Transportprozesse zwischen verschränkten Gefäßnetzwerken zum Ziel haben.:Introduction 1 1.1 Complex networks in biology . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.1 Flow networks in mammals . . . . . . . . . . . . . . . . . . . . 3 1.1.2 Network morphogenesis . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 State of the art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.2.1 Modelling flow network adaptation . . . . . . . . . . . . . . . . 8 1.2.2 Metrics for biological flow networks . . . . . . . . . . . . . . . . 11 Scaling in spatial networks . . . . . . . . . . . . . . . . . . . . . 12 Redundancy of flow networks . . . . . . . . . . . . . . . . . . . 13 1.3 Problem statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.3.1 Spatial embedding in metabolic costs models . . . . . . . . . . . 16 1.3.2 Characterizing three-dimensional reticulated networks . . . . . . 17 1.3.3 Optimal design for metabolite uptake . . . . . . . . . . . . . . . 20 2 Theory and Methods 23 2.1 Basic principles and mathematics . . . . . . . . . . . . . . . . . . . . 23 2.1.1 Mathematical basics . . . . . . . . . . . . . . . . . . . . . . . . 23 Linear equation systems . . . . . . . . . . . . . . . . . . . . . 23 Dynamical systems and optimization . . . . . . . . . . . . . . 25 Graph theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.1.2 Basic hydrodynamics . . . . . . . . . . . . . . . . . . . . . . . . 30 Momentum and mass balance . . . . . . . . . . . . . . . . . . . 30 Diffusion-Advection . . . . . . . . . . . . . . . . . . . . . . . . . 31 Flow in a thin channel . . . . . . . . . . . . . . . . . . . . . . . 32 2.1.3 Kirchhoff networks . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.2 Complex transport problems . . . . . . . . . . . . . . . . . . . . . . . . 36 2.2.1 Taylor dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.2.2 Flow-driven pruning . . . . . . . . . . . . . . . . . . . . . . . . 38 Metabolic cost functions . . . . . . . . . . . . . . . . . . . . . . 38 Adaptation and topological transitions . . . . . . . . . . . . . . 40 3 Results 43 3.1 On single network adaptation with fluctuating flow patterns . . . . . . 43 3.1.1 Incorporating flow fluctuations: Noisy, uncorrelated sink patterns 44 3.1.2 Fluctuation induced nullity transitions . . . . . . . . . . . . . . 48 3.1.3 Finite size effects and topological saturation limits . . . . . . . 52 3.2 On geometric coupling between intertwined networks . . . . . . . . . . 55 3.2.1 Power law model of interacting multilayer networks . . . . . . . 55 3.2.2 Adaptation dynamics of intertwined vessel systems . . . . . . . 57 x 3.2.3 Repulsive coupling induced nullity breakdown . . . . . . . . . . 59 3.2.4 Attractive coupling induced nullity onset . . . . . . . . . . . . 66 3.3 On generalizing and applying geometric laws to complex transport networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 3.3.1 Generalizing Murray’s law for complex flow networks . . . . . . 73 Murray’s law for fluctuating flows . . . . . . . . . . . . . . . . . 74 Murray’s Law for extended metabolic costs models . . . . . . . 77 3.3.2 Interpolating model parameters for intertwined networks . . . . 78 Testing ideal Kirchhoff networks . . . . . . . . . . . . . . . . . . 79 3.3.3 Identifying geometrical fingerprints in the liver lobule . . . . . . 85 3.4 On the optimization of metabolite uptake in complex flow networks . . 91 3.4.1 Metabolite transport in thin channel systems . . . . . . . . . . . 91 On single channel solutions . . . . . . . . . . . . . . . . . . . . 91 On detailed absorption rate models . . . . . . . . . . . . . . . . 93 On linear network solutions . . . . . . . . . . . . . . . . . . . . 96 On the uptake in spanning tree and reticulated networks . . . . 97 3.4.2 Optimizing metabolite uptake in shear-stress driven systems . . 100 Link-wise supply-demand model . . . . . . . . . . . . . . . . . . 101 Volume-wise supply-demand model . . . . . . . . . . . . . . . . 110 4 Discussion and Outlook 119 4.1 Summary of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 4.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 4.3 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 4.3.1 Metabolite transport in the liver lobule . . . . . . . . . . . . . . 124 Expansion of the Ostrenko model . . . . . . . . . . . . . . . . . 124 Complex multi transport probems in biology . . . . . . . . . . . 127 4.3.2 Absorption rate optimization and microscopic elimination models 128 Appendix A More on coupled intertwined networks 131 A.1 Coupling of Diamond lattices . . . . . . . . . . . . . . . . . . . . . . . 131 A.1.1 Repulsive coupling . . . . . . . . . . . . . . . . . . . . . . . . . 131 A.1.2 Attractive coupling . . . . . . . . . . . . . . . . . . . . . . . . . 133 A.2 Coupling of Laves Graphs . . . . . . . . . . . . . . . . . . . . . . . . . 134 A.2.1 Repulsive coupling . . . . . . . . . . . . . . . . . . . . . . . . . 134 A.2.2 Attractive coupling . . . . . . . . . . . . . . . . . . . . . . . . . 136 B More on metabolite uptake adaptation 139 B.1 Deriving dynamical systems from demand-supply relationships . . . . . 139 B.2 Microscopic uptake models . . . . . . . . . . . . . . . . . . . . . . . . . 142 B.2.1 Detailed uptake estimation in single layer systems . . . . . . . . 142 B.2.2 Detailed uptake estimation in liver sinusoids . . . . . . . . . . . 143 B.3 Metabolite uptake in three-dimensional plexi . . . . . . . . . . . . . . . 145 B.3.1 Link-wise demand adaptation . . . . . . . . . . . . . . . . . . . 145 B.3.2 Volume-wise demand adaptation . . . . . . . . . . . . . . . . . . 150 Bibliography 155 / Understanding the transport of fluid in complex organisms has proven to be a key challenge not only in the medical and biological sciences, but in physics and network theory as well. This is even more so as biologically-inspired design principles have been increasing in popularity, reliably generating solutions to common theoretical and technical problems. On that note, vascular networks in mammalian organs display a magnificent level of self-organization, allowing them to develop and mature, yet miraculously orchestrate the correct transport of oxygen, water, blood etc. This is achieved by a dedicated biochemical feedback system, which is coupled to macroscopic stimuli, such as mechanical stresses. Another important constraint for the morphogenesis of flow networks is their environment, as these networks are spatially embedded. They are therefore exposed to significant constraints with regards to their scalability and dynamical behavior, which are not yet well understood. This thesis addresses the current challenges of network characterization and morphogenesis modeling for three-dimensional embedded networks. In order to derive proper maturation mechanisms, we propose a set of toy models for the creation of non-planar, entangled and reticulated networks. The key mechanisms we focus on in this thesis are flow fluctuation, coupling of pairing structures and metabolite uptake. We show that in accordance with previous theoretical approaches, fluctuation induced nullity can be formulated as a single parameter problem. We demonstrate that the reticulation transition follows a logarithmic law and find plexi with certain topologies to have limited nullity transitions, rendering such plexi intrinsically wasteful in terms of fluctuation generated reticulation. Moreover, we formulate a new coupling model for entangled adapting networks as an approach for vasculature found in the liver lobules, pancreas, kidneys etc. We discuss a model based on local, distance-dependent interactions between pairs of three-dimensional network skeletons. In doing so we find unprecedented delay and breakdown of the fluctuation induced nullity transition for repulsive interactions. In addition we find a new nullity transition emerging for attractive coupling. Next, we study how flow fluctuations and complex metabolic costs can be incorporated into Murray’s Law. Utilizing this law for interpolation, we are able to derive order of magnitude estimation for the parameters in liver networks, suggesting fluctuation driven adaptation to be the dominant factor. We also conclude that attractive coupling is a reasonable mechanism to account for the maintenance of entangled structures. We test optimal metabolite uptake in Kirchhoff networks by evaluating the impact of solute uptake driven dynamics relative to wall-shear stress driven adaptation. Here, we find that a nullity transition emerges in case of a dominant metabolite uptake machinery. In addition to that, we find re-entrant behavior in case of high absorption rates and discover a complex interaction between shear-stress generation and feedback. Nevertheless, we conclude that metabolite uptake optimization is not likely to occur due to radial adaptation alone. We suggest areas for further studies, which should consider absorption rate variation in order to account for realistic uptake profiles. In our outlook, we suggest a complex morphogenesis model for intertwined networks based on the results of this thesis.:Introduction 1 1.1 Complex networks in biology . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.1 Flow networks in mammals . . . . . . . . . . . . . . . . . . . . 3 1.1.2 Network morphogenesis . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 State of the art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.2.1 Modelling flow network adaptation . . . . . . . . . . . . . . . . 8 1.2.2 Metrics for biological flow networks . . . . . . . . . . . . . . . . 11 Scaling in spatial networks . . . . . . . . . . . . . . . . . . . . . 12 Redundancy of flow networks . . . . . . . . . . . . . . . . . . . 13 1.3 Problem statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.3.1 Spatial embedding in metabolic costs models . . . . . . . . . . . 16 1.3.2 Characterizing three-dimensional reticulated networks . . . . . . 17 1.3.3 Optimal design for metabolite uptake . . . . . . . . . . . . . . . 20 2 Theory and Methods 23 2.1 Basic principles and mathematics . . . . . . . . . . . . . . . . . . . . 23 2.1.1 Mathematical basics . . . . . . . . . . . . . . . . . . . . . . . . 23 Linear equation systems . . . . . . . . . . . . . . . . . . . . . 23 Dynamical systems and optimization . . . . . . . . . . . . . . 25 Graph theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.1.2 Basic hydrodynamics . . . . . . . . . . . . . . . . . . . . . . . . 30 Momentum and mass balance . . . . . . . . . . . . . . . . . . . 30 Diffusion-Advection . . . . . . . . . . . . . . . . . . . . . . . . . 31 Flow in a thin channel . . . . . . . . . . . . . . . . . . . . . . . 32 2.1.3 Kirchhoff networks . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.2 Complex transport problems . . . . . . . . . . . . . . . . . . . . . . . . 36 2.2.1 Taylor dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.2.2 Flow-driven pruning . . . . . . . . . . . . . . . . . . . . . . . . 38 Metabolic cost functions . . . . . . . . . . . . . . . . . . . . . . 38 Adaptation and topological transitions . . . . . . . . . . . . . . 40 3 Results 43 3.1 On single network adaptation with fluctuating flow patterns . . . . . . 43 3.1.1 Incorporating flow fluctuations: Noisy, uncorrelated sink patterns 44 3.1.2 Fluctuation induced nullity transitions . . . . . . . . . . . . . . 48 3.1.3 Finite size effects and topological saturation limits . . . . . . . 52 3.2 On geometric coupling between intertwined networks . . . . . . . . . . 55 3.2.1 Power law model of interacting multilayer networks . . . . . . . 55 3.2.2 Adaptation dynamics of intertwined vessel systems . . . . . . . 57 x 3.2.3 Repulsive coupling induced nullity breakdown . . . . . . . . . . 59 3.2.4 Attractive coupling induced nullity onset . . . . . . . . . . . . 66 3.3 On generalizing and applying geometric laws to complex transport networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 3.3.1 Generalizing Murray’s law for complex flow networks . . . . . . 73 Murray’s law for fluctuating flows . . . . . . . . . . . . . . . . . 74 Murray’s Law for extended metabolic costs models . . . . . . . 77 3.3.2 Interpolating model parameters for intertwined networks . . . . 78 Testing ideal Kirchhoff networks . . . . . . . . . . . . . . . . . . 79 3.3.3 Identifying geometrical fingerprints in the liver lobule . . . . . . 85 3.4 On the optimization of metabolite uptake in complex flow networks . . 91 3.4.1 Metabolite transport in thin channel systems . . . . . . . . . . . 91 On single channel solutions . . . . . . . . . . . . . . . . . . . . 91 On detailed absorption rate models . . . . . . . . . . . . . . . . 93 On linear network solutions . . . . . . . . . . . . . . . . . . . . 96 On the uptake in spanning tree and reticulated networks . . . . 97 3.4.2 Optimizing metabolite uptake in shear-stress driven systems . . 100 Link-wise supply-demand model . . . . . . . . . . . . . . . . . . 101 Volume-wise supply-demand model . . . . . . . . . . . . . . . . 110 4 Discussion and Outlook 119 4.1 Summary of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 4.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 4.3 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 4.3.1 Metabolite transport in the liver lobule . . . . . . . . . . . . . . 124 Expansion of the Ostrenko model . . . . . . . . . . . . . . . . . 124 Complex multi transport probems in biology . . . . . . . . . . . 127 4.3.2 Absorption rate optimization and microscopic elimination models 128 Appendix A More on coupled intertwined networks 131 A.1 Coupling of Diamond lattices . . . . . . . . . . . . . . . . . . . . . . . 131 A.1.1 Repulsive coupling . . . . . . . . . . . . . . . . . . . . . . . . . 131 A.1.2 Attractive coupling . . . . . . . . . . . . . . . . . . . . . . . . . 133 A.2 Coupling of Laves Graphs . . . . . . . . . . . . . . . . . . . . . . . . . 134 A.2.1 Repulsive coupling . . . . . . . . . . . . . . . . . . . . . . . . . 134 A.2.2 Attractive coupling . . . . . . . . . . . . . . . . . . . . . . . . . 136 B More on metabolite uptake adaptation 139 B.1 Deriving dynamical systems from demand-supply relationships . . . . . 139 B.2 Microscopic uptake models . . . . . . . . . . . . . . . . . . . . . . . . . 142 B.2.1 Detailed uptake estimation in single layer systems . . . . . . . . 142 B.2.2 Detailed uptake estimation in liver sinusoids . . . . . . . . . . . 143 B.3 Metabolite uptake in three-dimensional plexi . . . . . . . . . . . . . . . 145 B.3.1 Link-wise demand adaptation . . . . . . . . . . . . . . . . . . . 145 B.3.2 Volume-wise demand adaptation . . . . . . . . . . . . . . . . . . 150 Bibliography 155

Web data is characterized by a great structural diversity as well as frequent changes, which poses a great challenge for web applications based on that data. We want to address this problem by developing a schema-optional and flexible data model that supports the integration of heterogenous and volatile web data. Therefore, we want to rely on graph-based models that allow to incrementally extend the schema by various information and constraints. Inspired by the on-going web 2.0 trend, we want users to participate in the design and management of the schema. By incrementally adding structural information, users can enhance the schema to meet their very specific requirements.

Today, virtually every company world-wide is connected to the Internet. This wide-spread connectivity has given rise to sophisticated, targeted, Internet-based attacks. For example, between 2012 and 2013 security researchers counted an average of about 74 targeted attacks per day. These attacks are motivated by economical, financial, or political interests and commonly referred to as “Advanced Persistent Threat (APT)” attacks. Unfortunately, many of these attacks are successful and the adversaries manage to steal important data or disrupt vital services. Victims are preferably companies from vital industries, such as banks, defense contractors, or power plants. Given that these industries are well-protected, often employing a team of security specialists, the question is: How can these attacks be so successful? Researchers have identified several properties of APT attacks which make them so efficient. First, they are adaptable. This means that they can change the way they attack and the tools they use for this purpose at any given moment in time. Second, they conceal their actions and communication by using encryption, for example. This renders many defense systems useless as they assume complete access to the actual communication content. Third, their actions are stealthy — either by keeping communication to the bare minimum or by mimicking legitimate users. This makes them “fly below the radar” of defense systems which check for anomalous communication. And finally, with the goal to increase their impact or monetisation prospects, their attacks are targeted against several companies from the same industry. Since months can pass between the first attack, its detection, and comprehensive analysis, it is often too late to deploy appropriate counter-measures at businesses peers. Instead, it is much more likely that they have already been attacked successfully. This thesis tries to answer the question whether the last property (industry-wide attacks) can be used to detect such attacks. It presents the design, implementation and evaluation of a community-based intrusion detection system, capable of protecting businesses at industry-scale. The contributions of this thesis are as follows. First, it presents a novel algorithm for community detection which can detect an industry (e.g., energy, financial, or defense industries) in Internet communication. Second, it demonstrates the design, implementation, and evaluation of a distributed graph mining engine that is able to scale with the throughput of the input data while maintaining an end-to-end latency for updates in the range of a few milliseconds. Third, it illustrates the usage of this engine to detect APT attacks against industries by analyzing IP flow information from an Internet service provider. Finally, it introduces a detection algorithm- and input-agnostic intrusion detection engine which supports not only intrusion detection on IP flow but any other intrusion detection algorithm and data-source as well.

Die sogenannten wunderbaren Modelle für Teilraumanordnungen, eingeführt von DeConcini und Procesi, basierend auf den Techniken der Fulton und MacPherson''schen Kompaktifzierung von Konfigurationsräumen, ermöglichen es, eine Fortsetzung von Feynmandistributionen auf die ihnen zugeordneten divergenten Teilräume in kanonischer Weise zu definieren. Dies wurde in der Dissertation von Christoph Bergbauer ausgearbeitet und diese Arbeit führt die dort präsentierten Ideen weiter aus. Im Unterschied formulieren wir die zentralen Begriffe nicht in geometrischer Sprache, sondern mit Hilfe der partiell geordneten Menge der divergenten Subgraphen eines Feynmangraphen. Dieser Ansatz ist inspiriert durch Feichtners Formulierung der wunderbaren Modellkonstruktion aus kombinatorischer Sicht. Diese Betrachtungsweise vereinfacht die Darstellung deutlich und führt zu einem besseren Verständnis der Fortsetzungs- bzw. Renormierungsoperatoren. Darüber hinaus erlaubt sie das Studium der Renormierungsgruppe, d.h. zu untersuchen, wie sich die renormierten Distributionen unter einem Wechsel des Renormierungspunktes verhalten. Wir zeigen, dass eine sogenannte endliche Renormierung sich darstellen läßt als eine Summe von durch die divergenten Subgraphen bestimmten Distributionen. Dies alles unterstreicht den wohlbekannten Fakt, dass perturbative Renormierung zum größten Teil durch die Kombinatorik von Feynmangraphen bestimmt ist und die analytischen Aspekte nur eine untergeordnete Rolle spielen. / The so-called wonderful models of subspace arrangements, developed in by DeConcini and Procesi, based on Fulton and MacPherson''s seminal paper on a compactification of configuration space, serve as a systematic way to resolve the singularities of Feynman distributions and define in this way canonical renormalization operators. In this thesis we continue the work of Bergbauer where wonderful models were introduced to solve the renormalization problem in position space. In contrast to the exposition there, instead of the subspaces in the arrangement of divergent loci we use the poset of divergent subgraphs of a given Feynman graph as the main tool to describe the wonderful construction and the renormalization operators. This is based on a review article by Feichtner where wonderful models were studied from a purely combinatorial viewpoint. The main motivation for this approach is the fact that both, the renormalization process and the model construction, are governed by the combinatorics of this poset. Not only simplifies this the exposition considerably, but it also allows to study the renormalization operators in more detail. Moreover, we explore the renormalization group in this setting, i.e. we study how the renormalized distributions change if one varies the renormalization points. We show that a so-called finite renormalization is expressed as a sum of distributions determined by divergent subgraphs. The bottom line is that - as is well known, at the latest since the discovery of a Hopf algebra structure underlying renormalization - the whole process of perturbative renormalization is governed by the combinatorics of Feynman graphs while the calculus involved plays only a supporting role.

Objectives: One of the most prominent features of schizophrenia is relatively lower general cognitive ability (GCA). An emerging approach to understanding the roots of variation in GCA relies on network properties of the brain. In this multi-center study, we determined global characteristics of brain networks using graph theory and related these to GCA in healthy controls and individuals with schizophrenia. Methods: Participants (N = 116 controls, 80 patients with schizophrenia) were recruited from four sites. GCA was represented by the first principal component of a large battery of neurocognitive tests. Graph metrics were derived from diffusion-weighted imaging. Results: The global metrics of longer characteristic path length and reduced overall connectivity predicted lower GCA across groups, and group differences were noted for both variables. Measures of clustering, efficiency, and modularity did not differ across groups or predict GCA. Follow-up analyses investigated three topological types of connectivity—connections among high degree “rich club” nodes, “feeder” connections to these rich club nodes, and “local” connections not involving the rich club. Rich club and local connectivity predicted performance across groups. In a subsample (N = 101 controls, 56 patients), a genetic measure reflecting mutation load, based on rare copy number deletions, was associated with longer characteristic path length. Conclusions: Results highlight the importance of characteristic path lengths and rich club connectivity for GCA and provide no evidence for group differences in the relationships between graph metrics and GCA.

Partitionierung von Graphen ist im Allgemeinen sehr schwierig. Es stehen derzeit keine Algorithmen zur Verfügung, die ein allgemeines Partitionierungsproblem effizient lösen. Aus diesem Grund werden heuristische Ansätze verfolgt. Zur Analyse dieser Heuristiken ist man derzeit gezwungen zufällige Graphen zu Verwenden. Daten realer Graphen sind derzeit entweder nur sehr schwer zu erheben (z.B. Internetgraph), oder aus rechtlichen bzw. wirtschaftlichen Gründen nicht zugänglich (z.B. soziale Netzwerke). Die untersuchten Heuristiken liefern teilweise nur unter bestimmten Voraussetzungen Ergebnisse. Einige arbeiten lediglich auf einer eingeschränkten Menge von Graphen, andere benötigen zum Erkennen einer Partition einen mit der Knotenzahl steigenden Durchschnittsgrad der Knoten, z.B. [DHM04]. Der im Zuge dieser Arbeit erstmals implementierte Algorithmus aus [CGL07a] benötigt lediglich einen konstanten Durchschnittsgrad der Knoten um eine Partition des Graphen, wenn diese existiert, zu erkennen. Insbesondere muss dieser Durchschnittsgrad nicht mit der Knotenzahl steigen. Nach der Implementierung erfolgten Tests des Algorithmus an zufälligen Graphen. Diese Graphen entsprachen dem Gnp-Modell mit eingepflanzter Partition. Die untersuchten Clusterprobleme waren dabei große Schnitte, kleine Schnitte und unabhängige Mengen. Der von der Art des Clusterproblems abhängige Durchschnittsgrad wurde während der Tests bestimmt.

The MPI_Barrier-collective operation, as a part of the MPI-1.1 standard, is extremely important for all parallel applications using it. The latency of this operation increases the application run time and can not be overlaid. Thus, the whole MPI performance can be decreased by unsatisfactory barrier latency. The main goals of this work are to lower the barrier latency for InfiniBand networks by analyzing well known barrier algorithms with regards to their suitability within InfiniBand networks, to enhance the barrier operation by utilizing standard InfiniBand operations as much as possible, and to design a constant time barrier for InfiniBand with special hardware support. This partition into three main steps is retained throughout the whole thesis. The first part evaluates publicly known models and proposes a new more accurate model (LoP) for InfiniBand. All barrier algorithms are evaluated within the well known LogP and this new model. Two new algorithms which promise a better performance have been developed. A constant time barrier integrated into InfiniBand as well as a cheap separate barrier network is proposed in the hardware section. All results have been implemented inside the Open MPI framework. This work led to three new Open MPI collective modules. The first one implements different barrier algorithms which are dynamically benchmarked and selected during the startup phase to maximize the performance. The second one offers a special barrier implementation for InfiniBand with RDMA and performs up to 40% better than the best solution that has been published so far. The third implementation offers a constant time barrier in a separate network, leveraging commodity components, with a latency of only 2.5 microseconds. All components have their specialty and can be used to enhance the barrier performance significantly.

Today, virtually every company world-wide is connected to the Internet. This wide-spread connectivity has given rise to sophisticated, targeted, Internet-based attacks. For example, between 2012 and 2013 security researchers counted an average of about 74 targeted attacks per day. These attacks are motivated by economical, financial, or political interests and commonly referred to as “Advanced Persistent Threat (APT)” attacks. Unfortunately, many of these attacks are successful and the adversaries manage to steal important data or disrupt vital services. Victims are preferably companies from vital industries, such as banks, defense contractors, or power plants. Given that these industries are well-protected, often employing a team of security specialists, the question is: How can these attacks be so successful? Researchers have identified several properties of APT attacks which make them so efficient. First, they are adaptable. This means that they can change the way they attack and the tools they use for this purpose at any given moment in time. Second, they conceal their actions and communication by using encryption, for example. This renders many defense systems useless as they assume complete access to the actual communication content. Third, their actions are stealthy — either by keeping communication to the bare minimum or by mimicking legitimate users. This makes them “fly below the radar” of defense systems which check for anomalous communication. And finally, with the goal to increase their impact or monetisation prospects, their attacks are targeted against several companies from the same industry. Since months can pass between the first attack, its detection, and comprehensive analysis, it is often too late to deploy appropriate counter-measures at businesses peers. Instead, it is much more likely that they have already been attacked successfully. This thesis tries to answer the question whether the last property (industry-wide attacks) can be used to detect such attacks. It presents the design, implementation and evaluation of a community-based intrusion detection system, capable of protecting businesses at industry-scale. The contributions of this thesis are as follows. First, it presents a novel algorithm for community detection which can detect an industry (e.g., energy, financial, or defense industries) in Internet communication. Second, it demonstrates the design, implementation, and evaluation of a distributed graph mining engine that is able to scale with the throughput of the input data while maintaining an end-to-end latency for updates in the range of a few milliseconds. Third, it illustrates the usage of this engine to detect APT attacks against industries by analyzing IP flow information from an Internet service provider. Finally, it introduces a detection algorithm- and input-agnostic intrusion detection engine which supports not only intrusion detection on IP flow but any other intrusion detection algorithm and data-source as well.

My thesis is mostly concerned with algebraic and combinatorial aspects of the theory of hyperplane arrangements. More specifically, I study the Orlik-Terao algebra of a hyperplane arrangement and the broken circuit complex of a matroid. The Orlik-Terao algebra is a useful tool for studying hyperplane arrangements, especially for characterizing some non-combinatorial properties. The broken circuit complex, on the one hand, is closely related to the Orlik-Terao algebra, and on the other hand, plays a crucial role in the study of many combinatorial problem: the coefficients of the characteristic polynomial of a matroid are encoded in the f-vector of the broken circuit complex of the matroid. Among main results of the thesis are characterizations of the complete intersection and Gorenstein properties of the broken circuit complex and the Orlik-Terao algebra. I also study the h-vector of the broken circuit complex of a series-parallel network and relate certain entries of that vector to ear decompositions of the network. An application of the Orlik-Terao algebra in studying the relation space of a hyperplane arrangement is also included in the thesis.

The interaction between the study of geometric and analytic aspects of Riemannian manifolds and that of graphs is a very amazing subject. The study of synthetic curvature notions on graphs adds new contributions to this topic. In this thesis, we mainly study two kinds of synthetic curvature notions: the Ollivier-Ricci cuvature on locally finite graphs and the combinatorial curvature on infinite semiplanar graphs. In the first part, we study the Ollivier-Ricci curvature. As known in Riemannian geometry, a lower Ricci curvature bound prevents geodesics from diverging too fast on average. We translate this Riemannian idea into a combinatorial setting using the Olliver-Ricci curvature notion. Note that on a graph, the analogue of geodesics starting in different directions, but eventually approaching each other again, would be a triangle. We derive lower and upper Ollivier-Ricci curvature bounds on graphs in terms of number of triangles, which is sharp for instance for complete graphs. We then describe the relation between Ollivier-Ricci curvature and the local clustering coefficient, which is an important concept in network analysis introduced by Watts-Strogatz. Furthermore, positive lower boundedness of Ollivier-Ricci curvature for neighboring vertices imply the existence of at least one triangle. It turns out that the existence of triangles can also improve Lin-Yau\'s curvature dimension inequality on graphs and then produce an implication from Ollivier-Ricci curvature lower boundedness to the curvature dimension inequality. The existence of triangles prevents a graph from being bipartite. A finite graph is bipartite if and only if its largest eigenvalue equals 2. Therefore it is natural that Ollivier-Ricci curvature is closely related to the largest eigenvalue estimates. We combine Ollivier-Ricci curvature notion with the neighborhood graph method developed by Bauer-Jost to study the spectrum estimates of a finite graph. We can always obtain nontrivial estimates on a non-bipartite graph even if its curvature is nonpositive. This answers one of Ollivier\'s open problem in the finite graph setting. In the second part of this thesis, we study systematically infinite semiplanar graphs with nonnegative combinatorial curvature. Unlike the previous Gauss-Bonnet formula approach, we explore an Alexandrov approach based on the observation that the nonnegative combinatorial curvature on a semiplanar graph is equivalent to nonnegative Alexandrov curvature on the surface obtained by replacing each face by a regular polygon of side length one with the same facial degree and gluing the polygons along common edges. Applying Cheeger-Gromoll splitting theorem on the surface, we give a metric classification of infinite semiplanar graphs with nonnegative curvature. We also construct the graphs embedded into the projective plane minus one point. Those constructions answer a question proposed by Chen. We further prove the volume doubling property and Poincare inequality which make the running of Nash-Moser iteration possible. We in particular explore the volume growth behavior on Archimedean tilings on a plane and prove that they satisfy a weak version of relative volume comparison with constant 1. With the above two basic inequalities in hand, we study the geometric function theory of infinite semiplanar graphs with nonnegative curvature. We obtain the Liouville type theorem for positive harmonic functions, the parabolicity. We also prove a dimension estimate for polynomial growth harmonic functions, which is an extension of the solution of Colding-Minicozzi of a conjecture of Yau in Riemannian geometry.