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n this paper we study invasion probabilities and invasion times of cooperative parasites spreading in spatially structured host populations. The spatial structure of the host population is given by a random geometric graph on [0,1]n, n∈N, with a Poisson(N)-distributed number of vertices and in which vertices are connected over an edge when they have a distance of at most rN∈Θ(Nβ−1n) for some 0<β<1 and N→∞. At a host infection many parasites are generated and parasites move along edges to neighbouring hosts. We assume that parasites have to cooperate to infect hosts, in the sense that at least two parasites need to attack a host simultaneously. We find lower and upper bounds on the invasion probability of the parasites in terms of survival probabilities of branching processes with cooperation. Furthermore, we characterize the asymptotic invasion time.
An important ingredient of the proofs is a comparison with infection dynamics of cooperative parasites in host populations structured according to a complete graph, i.e. in well-mixed host populations. For these infection processes we can show that invasion probabilities are asymptotically equal to survival probabilities of branching processes with cooperation.
Furthermore, we build in the proofs on techniques developed in [BP22], where an analogous invasion process has been studied for host populations structured according to a configuration model.
We substantiate our results with simulations.
The free energy of TAP-solutions for the SK-model of mean field spin glasses can be expressed as a nonlinear functional of local terms: we exploit this feature in order to contrive abstract REM-like models which we then solve by a classical large deviations treatment. This allows to identify the origin of the physically unsettling quadratic (in the inverse of temperature) correction to the Parisi free energy for the SK-model, and formalizes the true cavity dynamics which acts on TAP-space, i.e. on the space of TAP-solutions. From a non-spin glass point of view, this work is the first in a series of refinements which addresses the stability of hierarchical structures in models of evolving populations.
The specific temporal evolution of bacterial and phage population sizes, in particular bacterial depletion and the emergence of a resistant bacterial population, can be seen as a kinetic fingerprint that depends on the manifold interactions of the specific phage–host pair during the course of infection. We have elaborated such a kinetic fingerprint for a human urinary tract Klebsiella pneumoniae isolate and its phage vB_KpnP_Lessing by a modeling approach based on data from in vitro co-culture. We found a faster depletion of the initially sensitive bacterial population than expected from simple mass action kinetics. A possible explanation for the rapid decline of the bacterial population is a synergistic interaction of phages which can be a favorable feature for phage therapies. In addition to this interaction characteristic, analysis of the kinetic fingerprint of this bacteria and phage combination revealed several relevant aspects of their population dynamics: A reduction of the bacterial concentration can be achieved only at high multiplicity of infection whereas bacterial extinction is hardly accomplished. Furthermore the binding affinity of the phage to bacteria is identified as one of the most crucial parameters for the reduction of the bacterial population size. Thus, kinetic fingerprinting can be used to infer phage–host interactions and to explore emergent dynamics which facilitates a rational design of phage therapies.
Algorithms for the Maximum Cardinality Matching Problem which greedily add edges to the solution enjoy great popularity. We systematically study strengths and limitations of such algorithms, in particular of those which consider node degree information to select the next edge. Concentrating on nodes of small degree is a promising approach: it was shown, experimentally and analytically, that very good approximate solutions are obtained for restricted classes of random graphs. Results achieved under these idealized conditions, however, remained unsupported by statements which depend on less optimistic assumptions.
The KarpSipser algorithm and 1-2-Greedy, which is a simplified variant of the well-known MinGreedy algorithm, proceed as follows. In each step, if a node of degree one (resp. at most two) exists, then an edge incident with a minimum degree node is picked, otherwise an arbitrary edge is added to the solution.
We analyze the approximation ratio of both algorithms on graphs of degree at most D. Families of graphs are known for which the expected approximation ratio converges to 1/2 as D grows to infinity, even if randomization against the worst case is used. If randomization is not allowed, then we show the following convergence to 1/2: the 1-2-Greedy algorithm achieves approximation ratio (D-1)/(2D-3); if the graph is bipartite, then the more restricted KarpSipser algorithm achieves the even stronger factor D/(2D-2). These guarantees set both algorithms apart from other famous matching heuristics like e.g. Greedy or MRG: these algorithms depend on randomization to break the 1/2-barrier even for paths with D=2. Moreover, for any D our guarantees are strictly larger than the best known bounds on the expected performance of the randomized variants of Greedy and MRG.
To investigate whether KarpSipser or 1-2-Greedy can be refined to achieve better performance, or be simplified without loss of approximation quality, we systematically study entire classes of deterministic greedy-like algorithms for matching. Therefore we employ the adaptive priority algorithm framework by Borodin, Nielsen, and Rackoff: in each round, an adaptive priority algorithm requests one or more edges by formulating their properties---like e.g. "is incident with a node of minimum degree"---and adds the received edges to the solution. No constraints on time and space usage are imposed, hence an adaptive priority algorithm is restricted only by its nature of picking edges in a greedy-like fashion. If an adaptive priority algorithm requests edges by processing degree information, then we show that it does not surpass the performance of KarpSipser: our D/(2D-2)-guarantee for bipartite graphs is tight and KarpSipser is optimal among all such "degree-sensitive" algorithms even though it uses degree information merely to detect degree-1 nodes. Moreover, we show that if degrees of both nodes of an edge may be processed, like e.g. the Double-MinGreedy algorithm does, then the performance of KarpSipser can only be increased marginally, if at all. Of special interest is the capability of requesting edges not only by specifying the degree of a node but additionally its set of neighbors. This enables an adaptive priority algorithm to "traverse" the input graph. We show that on general degree-bounded graphs no such algorithm can beat factor (D-1)/(2D-3). Hence our bound for 1-2-Greedy is tight and this algorithm performs optimally even though it ignores neighbor information. Furthermore, we show that an adaptive priority algorithm deteriorates to approximation ratio exactly 1/2 if it does not request small degree nodes. This tremendous decline of approximation quality happens for graphs on which 1-2-Greedy and KarpSipser perform optimally, namely paths with D=2. Consequently, requesting small degree nodes is vital to beat factor 1/2.
Summarizing, our results show that 1-2-Greedy and KarpSipser stand out from known and hypothetical algorithms as an intriguing combination of both approximation quality and conceptual simplicity.
Die letzten Jahrzehnte brachten einen enormen Zuwachs des Wissens und Verständnisses über die molekularen Prozesse des Lebens.Möglich wurde dieser Zuwachs durch die Entwicklung diverser Methoden, mit denen beispielsweise gezielt die Konzentration einzelner Stoffe gemessen werden kann oder gar alle anwesenden Metaboliten eines biologischen Systems erfasst werden können. Die großflächige Anwendung dieser Methoden führte zur Ansammlung vieler unterschiedlicher -om-Daten, wie zum Beispiel Metabolom-, Proteom- oder Transkriptoms-Datensätzen. Die Systembiologie greift auf solche Daten zurück, um mathematische Modelle biologischer Systeme zu erstellen, und ermöglicht so ein Studium biologischer Systeme auch außerhalb des Labors.
Für größere biologische Systeme stehen jedoch meistens nicht alle Informationen über Stoffkonzentrationen oder Reaktionsgeschwindigkeiten zur Verfügung, um eine quantitative Modellierung, also die Beschreibung von Änderungsraten kontinuierlicher Variablen, durchführen zu können. In einem solchen Fall wird auf Methoden der qualitativen Modellierung zurückgegriffen. Eine dieser Methoden sind die Petrinetze (PN), welche in den 1960er Jahren von Carl Adam Petri entwickelt wurden, um nebenläufige Prozesse im technischen Umfeld zu beschreiben. Seit Anfang der 1990er Jahre finden PN auch Anwendung in der Systembiologie, um zum Beispiel metabolische Systeme oder Signaltransduktionswege zu modellieren. Einer der Vorteile dieser Methode ist zudem, dass Modelle als qualitative Beschreibung des Systems begonnen werden können und im Laufe der Zeit um quantitative Beschreibungen ergänzt werden können.
Zur Modellierung und Analyse von PN existieren bereits viele Anwendungen. Da das Konzept der PN jedoch ursprünglich nicht für die Systembiologie entwickelt wurde und meist im technischen Bereich verwendet wird, existierten kaum Anwendungen, die für den Einsatz in der Systembiologie entwickelt wurden. Daher ist auch die Durchführung der für die Systembiologie entwickelten Analysemethoden für PN nicht mit diesen Anwendungen möglich. Die Motivation des ersten Teiles dieser Arbeit war daher, eine Anwendung zu schaffen, die speziell für die PN-Modellierung und Analyse in der Systembiologie gedacht ist, also in ihren Analysemethoden und ihrer Terminologie sich an den Bedürfnissen der Systembiologie orientiert. Zudem sollte die Anwendung den Anwender bei der Auswertung der Resultate der Analysemethoden visuell unterstützen, indem diese direkt visuell im Kontext des PN gesetzt werden. Da bei komplexeren PN die Resultate der Analysemethoden in ihrer Zahl drastisch anwachsen, wird eine solche Auswertung dieser notwendig. Aus dieser Motivation heraus entstand die Anwendung MonaLisa, dessen Implementierung und Funktionen im ersten Teil der vorliegenden Arbeit beschrieben werden. Neben den klassischen Analysemethoden für PN, wie den Transitions- und Platz-Invarianten, mit denen grundlegende funktionale Module innerhalb eines PN gefunden werden können, wurden weitere, meist durch die Systembiologie entwickelte, Analysemethoden implementiert. Dazu zählen zum Beispiel die Minimal Cut Sets, die Maximal Common Transitions Sets oder Knock-out-Analysen. Mit MonaLisa ist aber auch die Simulation des dynamischen Verhaltens des modellierten biologischen Systems möglich. Hierzu stehen sowohl deterministische als auch stochastische Verfahren, beispielsweise der Algorithmus von Gillespie zur Simulation chemischer Systeme, zur Verfügung. Für alle zur Verfügung gestellten Analysemethoden wird ebenfalls eine visuelle Repräsentation ihrer Resultate bereitgestellt. Im Falle der Invarianten werden deren Elemente beispielsweise in der Visualisierung des PN eingefärbt. Die Resultate der Simulationen oder der topologischen Analyse können durch verschiedene Graphen ausgewertet werden. Um eine Schnittstelle zu anderen Anwendungen zu schaffen, wurde für MonaLisa eine Unterstützung einiger gängiger Dateiformate der Systembiologie geschaffen, so z.B. für SBML und KGML.
Der zweite Teil der Arbeit beschäftigt sich mit der topologischen Analyse eines Datensatzes von 2641 Gesamtgenom Modellen aus der path2models-Datenbank. Diese Modelle wurden automatisiert aus dem vorhandenen Wissen der KEGG- und der MetaCyc-Datenbank erstellt. Die Analyse der topologischen Eigenschaften eines Graphen ermöglicht es, grundlegende Aussagen über die globalen Eigenschaften des modellierten Systems und dessen Entstehungsprozesses zu treffen. Daher ist eine solche Analyse oft der erste Schritt für das Verständnis eines komplexen biologischen Systems. Für die Analyse der Knotengrade aller Reaktionen und Metaboliten dieser Modelle wurden sie in einem ersten Schritt in PN transformiert. Die topologischen Eigenschaften von metabolischen Systemen werden in der Literatur schon sehr gut beschrieben, wobei die Untersuchungen meist auf einem Netzwerk der Metaboliten oder der Reaktionen basieren. Durch die Verwendung von PN wird es möglich, die topologischen Eigenschaften von Metaboliten und Reaktionen in einem gemeinsamen Netzwerk zu untersuchen. Die Motivation hinter diesen Untersuchungen war, zu überprüfen, ob die schon beschriebenen Eigenschaften auch für eine Darstellung als PN zutreffen und welche neuen Eigenschaften gefunden werden können. Untersucht wurden der Knotengrad und der Clusterkoeffizient der Modelle. Es wird gezeigt, dass einige wenige Metaboliten mit sehr hohem Knotengrad für eine ganze Reihe von Effekten verantwortlich sind, wie beispielsweise dass die Verteilung des Knotengrades und des Clusterkoeffizienten, im Bezug auf Metaboliten, skalenfrei sind und dass sie für die Vernetzung der Nachbarschaft von Reaktionen verantwortlich sind. Weiter wird gezeigt, dass die Größe eines Modelles Einfluss auf dessen topologische Eigenschaften hat. So steigt die Vernetzung der Nachbarschaft eines Metaboliten, je mehr Metaboliten in einem biologischen System vorhanden sind, gleiches gilt für den durchschnittlichen Knotengrad der Metaboliten.
The behaviour of electronic circuits is influenced by ageing effects. Modelling the behaviour of circuits is a standard approach for the design of faster, smaller, more reliable and more robust systems. In this thesis, we propose a formalization of robustness that is derived from a failure model, which is based purely on the behavioural specification of a system. For a given specification, simulation can reveal if a system does not comply with a specification, and thus provide a failure model. Ageing usually works against the specified properties, and ageing models can be incorporated to quantify the impact on specification violations, failures and robustness. We study ageing effects in the context of analogue circuits. Here, models must factor in infinitely many circuit states. Ageing effects have a cause and an impact that require models. On both these ends, the circuit state is highly relevant, an must be factored in. For example, static empirical models for ageing effects are not valid in many cases, because the assumed operating states do not agree with the circuit simulation results. This thesis identifies essential properties of ageing effects and we argue that they need to be taken into account for modelling the interrelation of cause and impact. These properties include frequency dependence, monotonicity, memory and relaxation mechanisms as well as control by arbitrary shaped stress levels. Starting from decay processes, we define a class of ageing models that fits these requirements well while remaining arithmetically accessible by means of a simple structure.
Modeling ageing effects in semiconductor circuits becomes more relevant with higher integration and smaller structure sizes. With respect to miniaturization, digital systems are ahead of analogue systems, and similarly ageing models predominantly focus on digital applications. In the digital domain, the signal levels are either on or off or switching in between. Given an ageing model as a physical effect bound to signal levels, ageing models for components and whole systems can be inferred by means of average operation modes and cycle counts. Functional and faithful ageing effect models for analogue components often require a more fine-grained characterization for physical processes. Here, signal levels can take arbitrary values, to begin with. Such fine-grained, physically inspired ageing models do not scale for larger applications and are hard to simulate in reasonable time. To close the gap between physical processes and system level ageing simulation, we propose a data based modelling strategy, according to which measurement data is turned into ageing models for analogue applications. Ageing data is a set of pairs of stress patterns and the corresponding parameter deviations. Assuming additional properties, such as monotonicity or frequency independence, learning algorithm can find a complete model that is consistent with the data set. These ageing effect models decompose into a controlling stress level, an ageing process, and a parameter that depends on the state of this process. Using this representation, we are able to embed a wide range of ageing effects into behavioural models for circuit components. Based on the developed modelling techniques, we introduce a novel model for the BTI effect, an ageing effect that permits relaxation. In the following, a transistor level ageing model for BTI that targets analogue circuits is proposed. Similarly, we demonstrate how ageing data from analogue transistor level circuit models lift to purely behavioural block models. With this, we are the first to present a data based hierarchical ageing modeling scheme. An ageing simulator for circuits or system level models computes long term transients, solutions of a differential equation. Long term transients are often close to quasi-periodic, in some sense repetitive. If the evaluation of ageing models under quasi-periodic conditions can be done efficiently, long term simulation becomes practical. We describe an adaptive two-time simulation algorithm that basically skips periods during simulation, advancing faster on a second time axis. The bottleneck of two-time simulation is the extrapolation through skipped frames. This involves both the evaluation of the ageing models and the consistency of the boundary conditions. We propose a simulator that computes long term transients exploiting the structure of the proposed ageing models. These models permit extrapolation of the ageing state by means of a locally equivalent stress, a sort of average stress level. This level can be computed efficiently and also gives rise to a dynamic step control mechanism. Ageing simulation has a wide range of applications. This thesis vastly improves the applicability of ageing simulation for analogue circuits in terms of modelling and efficiency. An ageing effect model that is a part of a circuit component model accounts for parametric drift that is directly related to the operation mode. For example asymmetric load on a comparator or power-stage may lead to offset drift, which is not an empiric effect. Monitor circuits can report such effects during operation, when they become significant. Simulating the behaviour of these monitors is important during their development. Ageing effects can be compensated using redundant parts, and annealing can revert broken components to functional. We show that such mechanisms can be simulated in place using our models and algorithms. The aim of automatized circuit synthesis is to create a circuit that implements a specification for a certain use case. Ageing simulation can identify candidates that are more reliable. Efficient ageing simulation allows to factor in various operation modes and helps refining the selection. Using long term ageing simulation, we have analysed the fitness of a set of synthesized operational amplifiers with similar properties concerning various use cases. This procedure enables the selection of the most ageing resilient implementation automatically.
Considered are the classes QL (quasilinear) and NQL (nondet quasllmear) of all those problems that can be solved by deterministic (nondetermlnlsttc, respectively) Turmg machines in time O(n(log n) ~) for some k Effloent algorithms have time bounds of th~s type, it is argued. Many of the "exhausUve search" type problems such as satlsflablhty and colorabdlty are complete in NQL with respect to reductions that take O(n(log n) k) steps This lmphes that QL = NQL iff satisfiabdlty is m QL CR CATEGORIES: 5.25