<p>Evolutionary dynamics can be notoriously complex and difficult to analyze. In this dissertation I describe a population genetic regime where the dynamics are simple enough to allow a relatively complete and elegant treatment. Consider a haploid, asexual population, where each possible genotype has been assigned a fitness. When mutations enter a population sufficiently rarely, we can model the evolution of this population as a Markov chain where the population jumps from one genotype to another at the birth of each new mutant destined for fixation. Furthermore, if the mutation rates are assigned in such a manner that the Markov chain is reversible when all genotypes are assigned the same fitness, then it is still reversible when genotypes are assigned differing fitnesses. </p><p>The key insight is that this Markov chain can be analyzed using the spectral theory of finite-state, reversible Markov chains. I describe the spectral decomposition of the transition matrix and use it to build a general framework with which I address a variety of both classical and novel topics. These topics include a method for creating low-dimensional visualizations of fitness landscapes; a measure of how easy it is for the evolutionary process to `find' a specific genotype or phenotype; the index of dispersion of the molecular clock and its generalizations; a definition for the neighborhood of a genotype based on evolutionary dynamics; and the expected fitness and number of substitutions that have occurred given that a population has been evolving on the fitness landscape for a given period of time. I apply these various analyses to both a simple one-codon fitness landscape and to a large neutral network derived from computational RNA secondary structure predictions.</p> / Dissertation

Analysis of large networks in biology, science, technology and social systems have become very popular recently. These networks are mathematically represented as graphs. The task is then to extract relevant qualitative information about the empirical networks from the analysis of these graphs. It was found that a graph can be conveniently represented by the spectrum of a suitable difference operator, the normalized graph Laplacian, which underlies diffusions and random walks on graphs. When applied to large networks, this requires computation of the spectrum of large matrices. The normalized Laplacian matrices representing large networks are usually sparse and unstructured. The thesis consists in a systematic evaluation of the available eigenvalue solvers for nonsymmetric large normalized Laplacian matrices describing directed graphs of empirical networks. The methods include several Krylov subspace algorithms like implicitly restarted Arnoldi method, Krylov-Schur method and Jacobi-Davidson methods which are freely available as standard packages written in MATLAB or SLEPc, in the library written C++. The normalized graph Laplacian as employed here is normalized such that its spectrum is confined to the range [0, 2]. The eigenvalue distribution plays an important role in network analysis. The numerical task is then to determine the whole spectrum with appropriate eigenvalue solvers. A comparison of the existing eigenvalue solvers is done with Paley digraphs with known eigenvalues and for citation networks in sizes 400, 1100 and 4500 by computing the residuals.

Military analysis is becoming more reliant on constructive simulations for campaign modeling. Requirements for force-level capabilities, distributed command and control architectures, network centric operations, and increased levels of systems and operational integration are straining the analysis tools of choice. The models constructed are becoming more complex, both in terms of their composition and their behavior. They are complex in their composition because they are constituted from a large number of entities that interact nonlinearly through non-trivial networks and in their behavior because they display emergent characteristics. The modeling and simulation paradigm of choice for analyzing these systems of systems has been agent-based modeling and simulation. This construct is the most capable in terms of the characteristics of complex systems that it can capture, but it is the most demanding to construct, execute, verify and validate. This thesis is focused around two objectives. The first is to study the possibility of being able to compare two or more large-scale system architectures' capabilities without resorting to full-scale agent-based modeling and simulation. The second objective is to support the quantitative identification of the critical systems that compose the large-scale system architecture. The second objective will be crucial in the cases where a constructive simulation is the only option to capture the required behaviors of the complex system being studied. The enablers for this thesis are network modeling, graph theory, and in particular, spectral graph theory. The first hypothesis, stemmed from the first objective, states that if the capability of an architecture can be described as a series of functional cycles through the systems that compose them, then a simple network modeling construct can be employed to compare the different architectures' capabilities. The objective led to the second hypothesis, which states that a ranking based on the spectral characteristics of the network of functional interactions indicates the most critical systems. If modeling effort is focused on these systems, then the modeler can obtain the maximum fidelity model for the minimum effort.

Depuis plusieurs années, l’aéronautique est confrontée à une mutation majeure due à l’émergence des matériaux composites. Ce changement, justifié par les excellentes propriétés mécaniques des matériaux composites et un gain de masse important, implique une révision complète des réseaux de retour de courant. Pour faciliter cette révision, la thèse propose de lier au travers de l’analyse spectrale des graphes les performances des réseaux électriques avec leur topologie. Deux objectifs couplés sont étudiés : un dimensionnement topologique visant un bon compromis masse/robustesse et une stratégie de surveillance de ces réseaux. / The principles of the electrical system design in future aircrafts have to be reconsidered due to the emergence of new composite materials. The use of these materials for the aircraft structure has indeed implied a complete revision of on-board current return path networks. To facilitate this revision, it is proposed to link through the spectral graph analysis the performances of electrical networks with their topology. The aim of this thesis is to give topological drivers that could help the aeronautical engineers during the design process and then to propose a monitoring methodology.

This thesis advances the theory of network specialization by characterizing the effect of network specialization on the eigenvectors of a network. We prove and provide explicit formulas for the eigenvectors of specialized graphs based on the eigenvectors of their parent graphs. The second portion of this thesis applies network specialization to learning problems. Our work focuses on training reservoir computers to mimic the Lorentz equations. We experiment with random graph, preferential attachment and small world topologies and demonstrate that the random removal of directed edges increases predictive capability of a reservoir topology. We then create a new network model by growing networks via targeted application of the specialization model. This is accomplished iteratively by selecting top preforming nodes within the reservoir computer and specializing them. Our generated topology out-preforms all other topologies on average.

A well-known result in spectral graph theory states that if a graph has an equitable partition then the eigenvalues of the associated divisor graph are a subset of the graph's eigenvalues. A natural question question is whether it is possible to recover the remaining eigenvalues of the graph. Here we show that if a graph has a Hermitian adjacency matrix then the spectrum of the graph can be decomposed into a collection of smaller graphs whose eigenvalues are collectively the remaining eigenvalues of the graph. This we refer to as a complete equitable decomposition of the graph.

Developing nonparametric statistical methods and inference procedures for high-dimensional large data have been a challenging frontier problem of statistics. To attack this problem, in recent years, a clear rising trend has been observed with a radically different viewpoint--``Graph-based Nonparametrics," which is the main research focus of this dissertation. The basic idea consists of two steps: (i) representation step: code the given data using graphs, (ii) analysis step: apply statistical methods on the graph-transformed problem to systematically tackle various types of data structures. Under this general framework, this dissertation develops two major research directions. Chapter 2—based on Mukhopadhyay and Wang (2019a)—introduces a new nonparametric method for high-dimensional k-sample comparison problem that is distribution-free, robust, and continues to work even when the dimension of the data is larger than the sample size. The proposed theory is based on modern LP-nonparametrics tools and unexplored connections with spectral graph theory. The key is to construct a specially-designed weighted graph from the data and to reformulate the k-sample problem into a community detection problem. The procedure is shown to possess various desirable properties along with a characteristic exploratory flavor that has practical consequences. The numerical examples show surprisingly well performance of our method under a broad range of realistic situations. Chapter 3—based on Mukhopadhyay and Wang (2019b)—revisits some foundational questions about network modeling that are still unsolved. In particular, we present unified statistical theory of the fundamental spectral graph methods (e.g., Laplacian, Modularity, Diffusion map, regularized Laplacian, Google PageRank model), which are often viewed as spectral heuristic-based empirical mystery facts. Despite half a century of research, this question has been one of the most formidable open issues, if not the core problem in modern network science. Our approach integrates modern nonparametric statistics, mathematical approximation theory (of integral equations), and computational harmonic analysis in a novel way to develop a theory that unifies and generalizes the existing paradigm. From a practical standpoint, it is shown that this perspective can provide adequate guidance for designing next-generation computational tools for large-scale problems. As an example, we have described the high-dimensional change-point detection problem. Chapter 4 discusses some further extensions and application of our methodologies to regularized spectral clustering and spatial graph regression problems. The dissertation concludes with the a discussion of two important areas of future studies. / Statistics

Additive manufacturing (AM) is a powerful emerging technology for fabrication of components with complex geometries using a variety of materials. However, despite promising potential, due to the complexity of the process dynamics, how to ensure product quality and consistency of AM parts efficiently during the process still remains challenging. Therefore, the objective of this dissertation is to develop effective methodologies for online automatic quality monitoring and improvement, i.e., to build a basis for smart additive manufacturing. The fast-growing sensor technology can easily generate a massive amount of real-time process data, which provides excellent opportunities to address the barriers of online quality assurance in AM through data-driven perspectives. Although this direction is very promising, the online sensing data typically have high dimensionality and complex inherent structure, which causes the tasks of real-time data-driven analytics and decision-making to be very challenging. To address these challenges, multiple data-driven approaches have been developed in this dissertation to achieve effective feature extraction, process modeling, and closed-loop quality control. These methods are successfully validated by a typical AM process, namely, fused filament fabrication (FFF). Specifically, four new methodologies are proposed and developed as listed below, (1) To capture the variation of hidden patterns in sensor signals, a feature extraction approach based on spectral graph theory is developed for defect detection in online quality monitoring of AM. The most informative feature is extracted and integrated with a statistical control chart, which can effectively detect the anomalies caused by cyber-physical attack. (2) To understand the underlying structure of high dimensional sensor data, an effective dimension reduction method based on an integrated manifold learning approach termed multi-kernel metric learning embedded isometric feature mapping (MKML-ISOMAP) is proposed for online process monitoring and defect diagnosis of AM. Based on the proposed method, process defects can be accurately identified by supervised classification algorithms. (3) To quantify the layer-wise quality correlation in AM by taking into consideration of reheating effects, a novel bilateral time series modeling approach termed extended autoregressive (EAR) model is proposed, which successfully correlates the quality characteristics of the current layer with not only past but also future layers. The resulting model is able to online predict the defects in a layer-wise manner. (4) To achieve online defect mitigation for AM process, a closed-loop quality control system is implemented using an image analysis-based proportional-integral-derivative (PID) controller, which can mitigate the defects by adaptively adjusting machine parameters during the printing process in a timely manner. By fully utilizing the online sensor data with innovative data analytics and closed-loop control approaches, the above-proposed methodologies are expected to have excellent performance in online quality assurance for AM. In addition, these methodologies are inherently integrated into a generic framework. Thus, they can be easily transformed for applications in other advanced manufacturing processes. / Doctor of Philosophy / Additive manufacturing (AM) technology is rapidly changing the industry; and online sensor-based data analytics is one of the most effective enabling techniques to further improve AM product quality. The objective of this dissertation is to develop methodologies for online quality assurance of AM processes using sensor technology, advanced data analytics, and closed-loop control. It aims to build a basis for the implementation of smart additive manufacturing. The proposed new methodologies in this dissertation are focused to address the quality issues in AM through effective feature extraction, advanced statistical modeling, and closed-loop control. To validate their effectiveness and efficiency, a widely used AM process, namely, fused filament fabrication (FFF), is selected as the experimental platform for testing and validation. The results demonstrate that the proposed methods are very promising to detect and mitigate quality defects during AM operations. Consequently, with the research outcome in this dissertation, our capability of online defect detection, diagnosis, and mitigation for the AM process is significantly improved. However, the future applications of the accomplished work in this dissertation are not just limited to AM. The developed generic methodological framework can be further extended to many other types of advanced manufacturing processes.

La maladie de Huntington est une maladie neurodégénérative héréditaire qui est devenue un modèle d'étude pour comprendre la physiopathologie des maladies du cerveau associées à la production de protéines mal conformées et à la neurodégénérescence. Bien que plusieurs mécanismes aient été mis en avant pour cette maladie, dont plusieurs seraient aussi impliqués dans des pathologies plus fréquentes comme la maladie d’Alzheimer ou la maladie de Parkinson, nous ne savons toujours pas quels sont les mécanismes ou les profils moléculaires qui déterminent fondamentalement la dynamique des processus de dysfonction et de dégénérescence neuronale dans cette maladie. De même, nous ne savons toujours pas comment le cerveau peut résister aussi longtemps à la production de protéines mal conformées, ce qui suggère en fait que ces protéines ne présentent qu’une toxicité modérée ou que le cerveau dispose d'une capacité de compensation et de résilience considérable. L'hypothèse de mon travail de thèse est que l'intégration de données génomiques et transcriptomiques au travers des modèles qui récapitulent différentes phases biologiques de la maladie de Huntington peut permettre de répondre à ces questions. Dans cette optique, l'utilisation des réseaux de gènes et la mise en application de concepts issus de la théorie des graphes sont particulièrement bien adaptés à l'intégration de données hétérogènes, au travers des modèles et au travers des espèces. Les résultats de mon travail suggèrent que l'altération précoce (avant les symptômes, avant la mort cellulaire) et éventuellement dès le développement cérébral) des grandes voies de développement et de maintenance neuronale, puis la persistance voire l'aggravation de ces effets, sont à la base des processus physiopathologiques qui conduisent à la dysfonction puis à la mort neuronale. Ces résultats permettent aussi de prioriser des gènes et de générer des hypothèses fortes sur les cibles thérapeutiques les plus intéressantes à étudier d'un point de vue expérimental. En conclusion, mes recherches ont un impact à la fois fondamental et translationnel sur l'étude de la maladie de Huntington, permettant de dégager des méthodes d'analyse et des hypothèses qui pourraient avoir valeur thérapeutique pour les maladies neurodégénératives en général. / Huntington’s disease is a hereditary neurodegenerative disease that has become a model to understand physiopathological mechanisms associated to misfolded proteins that ocurs in brain diseases. Despite exciting findings that have uncover pathological mechanisms occurring in this disease and that might also be relevant to Alzheimer’s disease and Parkinson’s disease, we still do not know yet which are the mechanisms and molecular profiles that rule the dynamic of neurodegenerative processes in Huntington’s disease. Also, we do not understand clearly how the brain resist over such a long time to misfolded proteins, which suggest that the toxicity of these proteins is mild, and that the brain have exceptional compensation capacities. My work is based on the hypothesis that integration of ‘omics’ data from models that depicts various stages of the disease might be able to give us clues to answer these questions. Within this framework, the use of network biology and graph theory concepts seems particularly well suited to help us integrate heterogeneous data across models and species. So far, the outcome of my work suggest that early, pre-symptomatic alterations of signaling pathways and cellular maintenance processes, and persistency and worthening of these phenomenon are at the basis of physiopathological processes that lead to neuronal dysfunction and death. These results might allow to prioritize targets and formulate new hypotheses that are interesting to further study and test experimentally. To conclude, this work shall have a fundamental and translational impact to the field of Huntington’s disease, by pinpointing methods and hypotheses that could be valuable in a therapeutic perspective.

La maladie de Huntington est une maladie neurodégénérative héréditaire qui est devenue un modèle d'étude pour comprendre la physiopathologie des maladies du cerveau associées à la production de protéines mal conformées et à la neurodégénérescence. Bien que plusieurs mécanismes aient été mis en avant pour cette maladie, dont plusieurs seraient aussi impliqués dans des pathologies plus fréquentes comme la maladie d’Alzheimer ou la maladie de Parkinson, nous ne savons toujours pas quels sont les mécanismes ou les profils moléculaires qui déterminent fondamentalement la dynamique des processus de dysfonction et de dégénérescence neuronale dans cette maladie. De même, nous ne savons toujours pas comment le cerveau peut résister aussi longtemps à la production de protéines mal conformées, ce qui suggère en fait que ces protéines ne présentent qu’une toxicité modérée ou que le cerveau dispose d'une capacité de compensation et de résilience considérable. L'hypothèse de mon travail de thèse est que l'intégration de données génomiques et transcriptomiques au travers des modèles qui récapitulent différentes phases biologiques de la maladie de Huntington peut permettre de répondre à ces questions. Dans cette optique, l'utilisation des réseaux de gènes et la mise en application de concepts issus de la théorie des graphes sont particulièrement bien adaptés à l'intégration de données hétérogènes, au travers des modèles et au travers des espèces. Les résultats de mon travail suggèrent que l'altération précoce (avant les symptômes, avant la mort cellulaire) et éventuellement dès le développement cérébral) des grandes voies de développement et de maintenance neuronale, puis la persistance voire l'aggravation de ces effets, sont à la base des processus physiopathologiques qui conduisent à la dysfonction puis à la mort neuronale. Ces résultats permettent aussi de prioriser des gènes et de générer des hypothèses fortes sur les cibles thérapeutiques les plus intéressantes à étudier d'un point de vue expérimental. En conclusion, mes recherches ont un impact à la fois fondamental et translationnel sur l'étude de la maladie de Huntington, permettant de dégager des méthodes d'analyse et des hypothèses qui pourraient avoir valeur thérapeutique pour les maladies neurodégénératives en général. / Huntington’s disease is a hereditary neurodegenerative disease that has become a model to understand physiopathological mechanisms associated to misfolded proteins that ocurs in brain diseases. Despite exciting findings that have uncover pathological mechanisms occurring in this disease and that might also be relevant to Alzheimer’s disease and Parkinson’s disease, we still do not know yet which are the mechanisms and molecular profiles that rule the dynamic of neurodegenerative processes in Huntington’s disease. Also, we do not understand clearly how the brain resist over such a long time to misfolded proteins, which suggest that the toxicity of these proteins is mild, and that the brain have exceptional compensation capacities. My work is based on the hypothesis that integration of ‘omics’ data from models that depicts various stages of the disease might be able to give us clues to answer these questions. Within this framework, the use of network biology and graph theory concepts seems particularly well suited to help us integrate heterogeneous data across models and species. So far, the outcome of my work suggest that early, pre-symptomatic alterations of signaling pathways and cellular maintenance processes, and persistency and worthening of these phenomenon are at the basis of physiopathological processes that lead to neuronal dysfunction and death. These results might allow to prioritize targets and formulate new hypotheses that are interesting to further study and test experimentally. To conclude, this work shall have a fundamental and translational impact to the field of Huntington’s disease, by pinpointing methods and hypotheses that could be valuable in a therapeutic perspective.