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Discrete exponential families have drawn a lot of attention in probability, statistics, and machine learning, both classically and in the recent literature. This thesis studies in depth two discrete exponential families of concrete interest, (i) Exponential Random Graph Models (ERGMs) and (ii) Ising Models. In the ERGM setting, this thesis consider a “degree corrected” version of standard ERGMs, and in the Ising model setting, this thesis focus on Ising models on dense regular graphs, both from the point of view of statistical inference. The first part of the thesis studies the problem of testing for sparse signals present on the vertices of ERGMs. It proposes computably efficient tests for a wide class of ERGMs. Focusing on the two star ERGM, it shows that the tests studied are “asymptotically efficient” in all parameter regimes except one, which is referred to as “critical point”. In the critical regime, it is shown that improved detection is possible. This shows that compared to the standard belief, in this setting dependence is actually beneficial to the inference problem. The main proof idea for analyzing the two star ERGM is a correlations estimate between degrees under local alternatives, which is possibly of independent interest. In the second part of the thesis, we derive the limit of experiments for a class of one parameter Ising models on dense regular graphs. In particular, we show that the limiting experiment is Gaussian in the “low temperature” regime, non Gaussian in the “critical” regime, and an infinite collection of Gaussians in the “high temperature” regime. We also derive the limiting distributions of commonlt studied estimators, and study limiting power for tests of hypothesis against contiguous alternatives (whose scaling changes across the regimes). To the best of our knowledge, this is the first attempt at establishing the classical limits of experiments for Ising models (and more generally, Markov random fields).

The intention with this thesis is to investigate a dimer in a spin chain. Inorder to do that, a model from Z2-gauge theory is taken as the theoretical motivation to construct a discrete lattice with Ising spin properties. A dimer is then allowed to exist indirectly in the empty space between sites. We choose to tackle the problem through a quantum mechanical approach in 1+1-dimensions, distancing ourselves from the original description in quantum field theory. The exposition begins by reviewing the spatial construction of the entire chain as well as its components, and ends with a discussion of time development where the main concern is dispersion in addition to reflection against a static charge.

In this dissertation, the critical scaling behavior of magnetic Ising models with long range interactions is studied. These long range interactions, when imposed in addition to interactions on a regular lattice, lead to small-world graphs. By using large-scale Monte Carlo simulations, together with finite-size scaling, the critical behavior of a number of different models is obtained. The Ising models studied in this dissertation include the z-model introduced by Scalettar, standard small-world bonds superimposed on a square lattice, and physical small-world bonds superimposed on a square lattice. From the scaling results of the Binder 4th order cumulant, the order parameter, and the susceptibility, the long-range interaction is found to drive the systems behavior from Ising-like to mean field, and drive the critical point to a higher temperature. It is concluded that with a large amount of strong long-range connections (compared to the interactions on regular lattices), so the long-range connection density is non-vanishing, systems have mean field behavior. With a weak interaction that vanishes for an infinite system size or for vanishing density of long-range connections the systems have Ising-like critical behavior. The crossover from Ising-like to meanield behavior due to weak long-range interactions for systems with a large amount of long-range connections is also discussed. These results provide further evidence to support the existence of physical (quasi-) small-world nanomaterials.

Titanium-7%wt-Aluminum (Ti-7Al) has been of interest to the aerospace industry owing to its good structural and thermal properties. However, extensive research is still needed to study the structural behavior and determine the material properties of Ti-7Al. The homogenized macro-scale material properties are directly related to the crystallographic structure at the micro-scale. Furthermore, microstructural uncertainties arising from experiments and computational methods propagate on the material properties used for designing aircraft components. Therefore, multi-scale modeling is employed to characterize the microstructural features of Ti-7Al and computationally predict the macro-scale material properties such as Young's modulus and yield strength using machine learning techniques. Investigation of microstructural features across large domains through experiments requires rigorous and tedious sample preparation procedures that often lead to material waste. Therefore, computational microstructure reconstruction methods that predict the large-scale evolution of microstructural topology given the small-scale experimental information are developed to minimize experimental cost and time. However, it is important to verify the synthetic microstructures with respect to the experimental data by characterizing microstructural features such as grain size and grain shape. While the relationship between homogenized material properties and grain sizes of microstructures is well-studied through the Hall-Petch effect, the influences of grain shapes, especially in complex additively manufactured microstructure topologies, are yet to be explored. Therefore, this work addresses the gap in the mathematical quantification of microstructural topology by developing measures for the computational characterization of microstructures. Moreover, the synthesized microstructures are modeled through crystal plasticity simulations to determine the material properties. However, such crystal plasticity simulations require significant computing times. In addition, the inherent uncertainty of experimental data is propagated on the material properties through the synthetic microstructure representations. Therefore, the aforementioned problems are addressed in this work by explicitly quantifying the microstructural topology and predicting the material properties and their variations through the development of surrogate models. Next, this work extends the proposed multi-scale models of microstructure-property relationships to magnetic materials to investigate the ferromagnetic-paramagnetic phase transition. Here, the same Ising model-based multi-scale approach used for microstructure reconstruction is implemented for investigating the ferromagnetic-paramagnetic phase transition of magnetic materials. The previous research on the magnetic phase transition problem neglects the effects of the long-range interactions between magnetic spins and external magnetic fields. Therefore, this study aims to build a multi-scale modeling environment that can quantify the large-scale interactions between magnetic spins and external fields. / Doctor of Philosophy / Titanium-Aluminum (Ti-Al) alloys are lightweight and temperature-resistant materials with a wide range of applications in aerospace systems. However, there is still a lack of thorough understanding of the microstructural behavior and mechanical performance of Titanium-7wt%-Aluminum (Ti-7Al), a candidate material for jet engine components. This work investigates the multi-scale mechanical behavior of Ti-7Al by computationally characterizing the micro-scale material features, such as crystallographic texture and grain topology. The small-scale experimental data of Ti-7Al is used to predict the large-scale spatial evolution of the microstructures, while the texture and grain topology is modeled using shape moment invariants. Moreover, the effects of the uncertainties, which may arise from measurement errors and algorithmic randomness, on the microstructural features are quantified through statistical parameters developed based on the shape moment invariants. A data-driven surrogate model is built to predict the homogenized mechanical properties and the associated uncertainty as a function of the microstructural texture and topology. Furthermore, the presented multi-scale modeling technique is applied to explore the ferromagnetic-paramagnetic phase transition of magnetic materials, which causes permanent failure of magneto-mechanical components used in aerospace systems. Accordingly, a computational solution is developed based on an Ising model that considers the long-range spin interactions in the presence of external magnetic fields.

In this work, we design parallel Monte Carlo algorithms for the Ising spin model on a hierarchical cluster. A hierarchical cluster can be considered as a cluster of homogeneous nodes which are partitioned into multiple supernodes such that communication across homogenous clusters is represented by a supernode topological network. We consider different data layouts and provide equations for choosing the best data layout under such a network paradigm. We show that the data layouts designed for a homogeneous cluster will not yield results as good as layouts designed for a hierarchical cluster. We derive theoretical results on the performance of the algorithms on a modified version of the LogP model that represents such tiered networking, and present simulation results to analyze the utility of the theoretical design and analysis. Furthermore, we consider the 3-D Ising model and design parallel algorithms for sweep spin selection on both homogeneous and hierarchical clusters. We also discuss the simulation of hierarchical clusters on a homogeneous set of machines, and the efficient implementation of the parallel Ising model on such clusters. / Master of Science

In this dissertation, we develop a model selection and estimation methodology for nonstationary spatial fields. Large, spatially correlated data often cover a vast geographical area. However, local spatial regions may have different mean and covariance structures. Our methodology accomplishes three goals: (1) cluster locations into small regions with distinct, stationary models, (2) perform Bayesian model selection within each cluster, and (3) correlate the model selection and estimation in nearby clusters. We utilize the Conditional Autoregressive (CAR) model and Ising distribution to provide intra-cluster correlation on the linear effects and model inclusion indicators, while modeling inter-cluster correlation with separate Gaussian processes. We apply our model selection methodology to a dataset involving the prediction of Brook trout presence in subwatersheds across Pennsylvania. We find that our methodology outperforms the stationary spatial model and that different regions in Pennsylvania are governed by separate Gaussian process regression models. / Ph. D. / In this dissertation, we develop a statistical methodology for analyzing data where observations are related to each other due to spatial proximity. Our overall goal is to determine which attributes are important when predicting the response of interest. However, the effect and importance of an attribute may differ depending on the spatial location of the observation. Our methodology accomplishes three goals: (1) partition the observations into small spatial regions, (2) determine which attributes are important within each region, and (3) enforce that the importance of variables should be similar in regions that are near each other. We apply our technique to a dataset involving the prediction of Brook trout presence in subwatersheds across Pennsylvania.

Neste trabalho estudamos a existência de ordem de longo alcance em modelos ferromagnéticos na presença de um campo externo cuja configuração apresenta um padrão tipicamente aleatório. Provamos por meio do argumento de Peierls modificado por Griffiths para o estudo de um antiferromagneto, que o modelo de Ising ferromagnético bidimensional exibe, para um campo alternado de intensidade fraca, ordem de longo alcance `a temperatura finita. Propomos dar um passo além considerando campos auto-similares esparsos, cuja soma é nula em todas as escalas. Estudamos também o modelo hierárquico em duas dimensões, para o qual provamos a existência de ordem de longo alcance a temperatura finita, na ausência de campo externo e para um campo com regiões irregulares esparsas. Provamos que os resultados do modelo de contornos hierárquicos são equivalentes aos resultados do modelo hierárquico em duas dimensões. Por fim, provamos através do método do limite infravermelho existência de ordem de longo alcance no modelo N-vetorial com campo alternado, de intensidade fraca, para d >= 3, sob a hipótese de que a variância do estado associado `a interação com o campo apresenta cardinalidade inferior a do volume do sistema. Mostramos, sob hipóteses similares, que o modelo N-vetorial hierárquico com campo externo, esparso e de intensidade pequena, apresenta ordem de longo alcance a baixas temperaturas. / In this work we study the existence of long range order for ferromagnetic models in the presence of an external field whose configuration has a pattern typically random. We prove, via the Peierls\' argument modified by Griffiths in his study of an antiferromagnet, that the two dimensional ferromagnetic Ising model for a staggered field exhibits long-range order at finite temperature and small field intensity. We propose to give a further step considering sparse self similar fields, whose sum is zero in all scales. We study as well the hierarchical model in two dimensions, where we prove existence of long-range order at finite temperature in the absence of external field and for a field configuration with sparse irregular regions. We prove that the results for the two-dimensional hierarchical contours model are equivalent to the results of the hierarchical model in two dimensions. Lastly, we prove via infrared bound method, existence of long range order in the N-vector model with a staggered and weak external field for d >= 3, under the hypothesis that the variance of the state connected with the field interaction has cardinality lower than volume. We show, under similar hypotheses, that the N-vector hierarchical model with a sparse field of low intensity has long range ordem at low temperatures.

O modelo de Ising dipolar bidimensional inclui, além da interação ferromagnética entre os primeiros vizinhos, interações de longo alcance entre os momentos de dipolo magnético dos spins. A presença da interação dipolar muda completamente o sistema, apresentando um rico diagrama de fase, cujas características têm originado inúmeros estudos na literatura. Além disso, a possibilidade de explicar fenômenos observados em filmes magnéticos ultrafinos, os quais possuem diversas aplicações em àreas tecnológicas, também motiva o estudo deste modelo. O estado fundamental ferromagnético do modelo de Ising puro é alterado para uma série de fases do tipo faixas, as quais consistem em domínios ferromagnéticos de largura $h$ com magnetizações opostas. A largura das faixas depende da razao $\\delta$ das intensidades dos acoplamentos ferromagnético e dipolar. Através de simulações de Monte Carlo e técnicas de repesagem em histogramas múltiplos identificamos as temperaturas críticas de tamanho finito para as transições de fase quando $\\delta=2$, o que corresponde a $h=2$. Calculamos o calor específico e a susceptibilidade do parâmetro de ordem, no intervalo de temperaturas onde as transições são observadas, para diferentes tamanhos de rede. As técnicas de repesagem permitem-nos explorar e identificar máximos distintos nessas funções da temperatura e, desse modo, estimar as temperaturas críticas de tamanho finito com grande precisão. Apresentamos evidências numéricas da existência de uma fase nemática de Ising para tamanhos grandes de rede. Em nossas simulações, observamos esta fase para tamanhos de rede a partir de $L=48$. Para verificar o quanto a interação dipolar de longo alcance afeta as estimativas físicas, nós calculamos o tempo de autocorrelação integrado nas séries temporais da energia. Inferimos daí quão severo é o critical slowing down (decaimento lento crítico) para esse sistema próximo às transições de fase termodinâmicas. Os resultados obtidos utilizando um algoritmo de atualização local foram comparados com os resultados obtidos utilizando o algoritmo multicanônico. / Two-dimensional spin model with nearest-neighbor ferromagnetic interaction and long-range dipolar interactions exhibit a rich phase diagram, whose characteristics have been exploited by several studies in the recent literature. Furthermore, the possibility of explain observed phenomena in ultrathin magnetic films, which have many technological applications, also motivates the study of this model. The presence of dipolar interaction term changes the ferromagnetic ground state expected for the pure Ising model to a series of striped phases, which consist of ferromagnetic domains of width $h$ with opposite magnetization. The width of the stripes depends on the ratio $\\delta$ of the ferromagnetic and dipolar couplings. Monte Carlo simulations and reweighting multiple histograms techniques allow us to identify the finite-size critical temperatures of the phase transitions when $\\delta=2$, which corresponds to $h=2$. We calculate, for different lattice sizes, the specific heat and susceptibility of the order parameter around the transition temperatures by means of reweighting techniques. This allows us to identify in these observables, as functions of temperature, the distinct maxima and thereby to estimate the finite-size critical temperatures with high precision. We present numerical evidence of the existence of a Ising nematic phase for large lattice sizes. Our results show that simulations need to be performed for lattice sizes at least as large as $L=48$ to clearly observe the Ising nematic phase. To access how the long-range dipolar interaction may affect physical estimates we also evaluate the integrated autocorrelation time in energy time series. This allows us to infer how severe is the critical slowing down for this system with long-range interaction and nearby thermodynamic phase transitions. The results obtained using a local update algorithm are compared with results obtained using the multicanonical algorithm.

Recent experimental techniques in biology made possible the acquisition of overwhelming amounts of data concerning complex biological networks, such as neural networks, gene regulation networks and protein-protein interaction networks. These techniques are able to record states of individual components of such networks (neurons, genes, proteins) for a large number of configurations. However, the most biologically relevantinformation lies in their connectivity and in the way their components interact, information that these techniques aren't able to record directly. The aim of this thesis is to study statistical methods for inferring information about the connectivity of complex networks starting from experimental data. The subject is approached from a statistical physics point of view drawing from the arsenal of methods developed in the study of spin glasses. Spin-glasses are prototypes of networks of discrete variables interacting in a complex way and are widely used to model biological networks. After an introduction of the models used and a discussion on the biological motivation of the thesis, all known methods of network inference are introduced and analysed from the point of view of their performance. Then, in the third part of the thesis, a new method is proposed which relies in the remark that the interactions in biology are not necessarily symmetric (i.e. the interaction from node A to node B is not the same as the one from B to A). It is shown that this assumption leads to methods that are both exact and efficient. This means that the interactions can be computed exactly, given a sufficient amount of data, and in a reasonable amount of time. This is an important original contribution since no other method is known to be both exact and efficient.