Dans cette thèse, nous nous intéressons à la monotonie de la vitesse de la marche aléatoire excitée (MAE) avec biais $bein[0,1]$ dans la première direction $e_1$. Nous présentons une nouvelle preuve de la monotonie de la vitesse pour des grandes dimensions $dgeq d_0$ et pour le cas où le paramètre $be$ est petit quand $dgeq 8$. Ensuite, nous considérons les marches aléatoires avec plusieurs cookies aléatoires. La monotonie de la vitesse est ausi prouvée pour les cas particuliers par exemple des dimensions sont grandes, le paramètre de dérive $be$ est petit ou le nombre de cookies est grand. Ce sont les cas où la marche aléatoire est proche à la marche aléatoire simple. Pour l'existence de la vitesse, nous avons montré la loi des grands nombres pour un cas particulier du cookie aléatoire stationaire, mais nous n'arrivons pas encore pour le cas stationaire. Sur la monotonie, nous avons aussi vérifié que le nombre de points visités par la marche aléatoire simple avec biais $be$ est croissant.Finalement, une question très interessant: la monotonie de la vitesse, est-elle vraie pour la MAE pour les petites dimensions $2leq dleq 8.$ Pour cette motivation, nous avons prouvé que la vitesse est indéfiniment différentiable pour $be>0.$ Au point critique $0$, nous avons prouvé que la dérivée de la vitesse existe et égale $0$ pour $d=2$, existe et est positive pour $dgeq 4.$ Mais nous ne savons pas encore si la dérivée de l'ordre 2 en point $0$ existe ou au moin la dérivée est continue en $0$ pour prouver la monotonie de la vitesse au voisinage de $0$? / In this thesis, we are interested in the monotonicity of the speed of the excited random walk (ERW) with bias $bein[0,1]$ in the first direction $e_1.$ The speed is defined as the limit obtained by the law of large number for the horizontal component. The speed depend on the bias $be.$ We present a new proof of the monotonicity of the speed for the dimension $dgeq d_0$, where $d_0$ is large enough, or for the parameter $be$ is small when $dgeq 8$. After that, we consider the random walk with multi-random cookies. The monotonicity of the speed is also proved for some particular cas, for exemple when the dimension is high, or the parameter drift is small, or the number of cookies is large. These are the cas where the walk is near the simple random walk. For the existence of the speed, we also proved the law of large number for a particular cas of stationary cookie but we haven't yet gotten the cas stationary. On the monotonicity, we also proved the rang of the simple random walk with drift $be$ is increasing in the drift. Finally, a question very interesting: the monotonicity of the speed of ERW is true for the small dimension $2leq dleq 8$, isn't it? For this motivation, we proved the speed is infinitly differentiable for all $be>0.$ At the critical point $0,$ we also proved the derivative of the speed at $0$ exists and equals $0$ for $d=2$, exists and is positive for $dgeq 4.$ But we haven't yet known if the derivative of order $2$ at $0$ exists or at least the derivative is continuous at $0$ to prove the monotonicity of the speed in a neighbor of $0$.

This work deals with the one-dimensional Stefan problem with a general time- dependent boundary condition at the fixed boundary. The solution will be obtained using a discrete random walk method and the results will be compared qualitatively with analytical- and finite difference method solutions. A critical part has been to model the moving boundary with the random walk method. The results show that the random walk method is competitive in relation to the finite difference method and has its advantages in generality and low effort to implement. The finite difference method has, on the other hand, higher accuracy for the same computational time with the here chosen step lengths. For the random walk method to increase the accuracy, longer execution times are required, but since the method is generally easily adapted for parallel computing, it is possible to speed up. Regarding applications for the Stefan problem, there are a large range of examples such as climate models, the diffusion of lithium-ions in lithium-ion batteries and modelling steam chambers for oil extraction using steam assisted gravity drainage.

Neste trabalho, estudamos o passeio aleatório do elefante introduzido em (SCHUTZ; TRIMPER, 2004). Um processo estocástico não Markoviano com memória de alcance ilimitada que apresenta transição de fase. Nosso objetivo é demonstrar a convergência quase certa do passeio aleatório do elefante nos casos subcrítico e crítico. Além destes resultado, também apresentamos a demonstração do Teorema Central do Limite para ambos os regimes. Para o caso supercrítico, vamos demonstrar a convergência do passeio aleatório do elefante para uma variável aleatória não normal com base nos artigos (BAUR; BERTOIN, 2016), (BERCU, 2018) e (COLETTI; GAVA; SCHUTZ, 2017b). / In this work we study the elephant random walk introduced in (SCHUTZ; TRIMPER, 2004), a discrete time, non-Markovian stochastic process with unlimited range memory that presents phase transition. Our objective is to proof the almost sure convergence for the subcritical and critical regimes of the model. We also present a demonstration of the Central Limit Theorem for both regimes. For the supercritical regime we proof the convergence of the elephant random walk to a non-normal random variable based on the articles (BAUR; BERTOIN, 2016), (BERCU, 2018) and (COLETTI; GAVA; SCHUTZ, 2017b).

Redes complexas têm sido cada vez mais utilizadas para a modelagem e análise dos mais diversos sistemas da natureza. Um dos tópicos mais estudados na área de redes está relacionado com a identificação e caracterização de grupos de nós mais conectados entre si do que com o restante da rede, chamados de comunidades. Neste trabalho, mostramos que comunidades podem ser caracterizadas por quatro classes gerais de propriedades, relacionadas com a topologia interna, dinâmica interna, fronteira topológica, e fronteira dinâmica das comunidades. Verificamos como estas diferentes características influenciam em dinâmicas ocorrendo sobre a rede. Em especial, estudamos o inter-relacionamento entre a topologia e a dinâmica das comunidades para cada uma dessas quatro classes de atributos. Mostramos que certas propriedades provocam a alteração desse inter-relacionamento, dando origem ao que chamamos de comportamento específico de comunidades. De forma a apresentarmos e analisarmos este conceito nos quatro casos considerados, estudamos as seguintes combinações topológicas e dinâmicas. Na primeira, investigamos o passeio aleatório tradicional ocorrendo sobre redes direcionadas, onde mostramos que a direção das conexões entre comunidades é o principal fator de alteração no relacionamento topologia-dinâmica. Aplicamos a metodologia proposta em uma rede real, definida por módulos corticais de animais do gênero Macaca. O segundo caso estudado aborda o passeio aleatório enviesado ocorrendo sobre redes não direcionadas. Mostramos que o viés associado às transições da dinâmica se tornam cada vez mais relevantes com o aumento da modularidade da rede. Verificamos também que a descrição da dinâmica a nível de comunidades possibilita modelarmos com boa acurácia o fluxo de passageiros em aeroportos. A terceira análise realizada envolve a dinâmica neuronal integra-e-dispara ocorrendo sobre comunidades geradas segundo o modelo Watts-Strogatz. Mostramos que as comunidades podem possuir não apenas diferentes níveis de ativação dinâmica, como também apresentar diferentes regularidades de sinal dependendo do parâmetro de reconexão utilizado na criação das comunidades. Por último, estudamos a influência das posições de conexões inibitórias na dinâmica integra-e-dispara, onde mostramos que a inibição entre comunidades dá origem a interessantes variações na ativação global da rede. As análises realizadas revelam a importância de, ao modelarmos sistemas reais utilizando redes complexas, considerarmos alterações de parâmetros do modelo na escala de comunidades. / There has been a growing interest in modeling diverse types of real-world systems through the tools provided by complex network theory. One of the main topics of research in this area is related to the identification and characterization of groups, or communities, of nodes more densely connected between themselves than with the rest of the network. We show that communities can be characterized by four general classes of features, associated with the internal topology, internal dynamics, topological border, and dynamical border of the communities. We verify that these characteristics have direct influence on the dynamics taking place over the network. Particularly, for each considered class we study the interdependence between the topology and the dynamics associated with each network community. We show that some of the studied properties can influence the topology-dynamics interdependence, inducing what we call the communities specific behavior. In order to present and characterize this concept on the four considered classes, we study the following combinations of network topology and dynamics. We first investigate traditional random walks taking place on a directed network. We demonstrate that, for this dynamics, the direction of the edges between communities represents the main method for the modification of the topology-dynamics relationship. We apply the developed approach on a real-world network, defined by the connectivity between cortical regions in primates of the Macaca genus. The second studied case considers the biased random walk on undirected networks. We demonstrate that the transition bias of this dynamics becomes more relevant for higher network modularity. In addition, we show that the biased random walk can be used to model with good accuracy the passenger flow inside the communities of two airport networks. The third analysis is done on a neuronal dynamics, called integrate-and-fire, applied to networks composed of communities generated by the Watts-Strogatz model. We show that the considered communities can not only posses distinct dynamical activation levels, but also yield different signal regularity. Lastly, we study the influence of the positions of inhibitory connections on the integrate-and-fire dynamics. We show that inhibitory connections placed between communities can have a non-trivial influence on the global behavior of the dynamics. The current study reveals the importance of considering parameter variations of network models at the scale of communities.

Na natureza e na ciência, dados e informações que desviam significativamente da média frequentemente possuem grande relevância. Esses dados são usualmente denominados na literatura como outliers. A identificação de outliers é importante em muitas aplicações reais, tais como detecção de fraudes, diagnóstico de falhas, e monitoramento de condições médicas. Nos últimos anos tem-se testemunhado um grande interesse na área de Redes Complexas. Redes complexas são grafos de grande escala que possuem padrões de conexão não trivial, mostrando-se uma poderosa maneira de representação e abstração de dados. Embora um grande montante de resultados tenham sido reportados nesta área de pesquisa, pouco tem sido explorado acerca de detecção de outliers em redes complexas. Considerando-se a dinâmica de uma caminhada aleatória, foram propostos neste trabalho uma medida de distância e um método de ranqueamento de outliers. Através desta técnica, é possível detectar como outlier não somente nós periféricos, mas também nós centrais (hubs), depedendo da estrutura da rede. Também foi identificado que existem características bem definidas entre os nós outliers, relacionadas a funcionalidade dos mesmos para a rede. Além disso, foi descoberto que nós outliers têm papel importante para a rotulação a priori na tarefa de detecção de comunidades semi-supervisionada. Isto porque os nós centrais são bons difusores de informação e os nós periféricos encontram-se em regiões de borda de comunidade. Baseado nessa observação, foi proposto um método de detecção de comunidades semi-supervisionado. Os resultados de simulações mostram que essa abordagem é promissora / In nature and science, information and data that deviate significantly from the average value often have great relevance. These data are often called in literature as outliers. Outlier identification is important in many real applications, such as fraud detection, fault diagnosis, monitoring of medical conditions. In recent years, it has been witnessed a great interest in the area of Complex Networks. Complex networks are large-scale graphs with non-trivial connection patterns, proving to be a powerful way of data representation and abstraction. Although a large amount of results have been reported in this research area, little has been explored about the outlier detection in complex networks. Considering the dynamics of a random walk, we proposed in this paper a distance measure and a outlier ranking method. By using this technique, we can detect not only peripheral nodes, but also central nodes (hubs) as outliers, depending on the network structure. We also identified that there are well defined relationship between the outlier nodes and the functionality of the same nodes for the network. Furthermore, we found that outliers play an important role to label a priori nodes in the task of semi-supervised community detection. This is because the hubs are good information disseminators and peripheral nodes are usually localized in the regions of community edges. Based on this observation, we proposed a method of semi-supervised community detection. The simulation results show that this approach is promising

Made available in DSpace on 2017-07-21T19:26:05Z (GMT). No. of bitstreams: 1 Marily Aparecida Benicio.pdf: 2662358 bytes, checksum: ba11ae50b21bd9be5feba0d2bf5fd563 (MD5) Previous issue date: 2013-04-09 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Studies of complex networks help us to understand and model many real world situations. The world abounds in networks that can be found in many real contexts. The term network refers to relations between two sets and can be represented by means of graph theory. The classification of complex networks is given according to the models created to represent them, such as Random networks, networks of Small World, No Scaling networks and hierarchical networks. From the perspective of complex networks, a study which make significant contributions analysis is the phenomenon of diffusion of information in networks, which can be understood through the random walk process, which is characterized by a stochastic used as a mechanism transportation and research in complex networks. A random walk in complex networks can be used to check the behavior of each network model front dissipation. Each network model presents a different behavior with respect to the number of random walkers that pass through the network node over time. The number of walkers will depend on the structure of the networks generated by each model and measures of centrality of each node. Measures of centrality of the vertices of the network are useful for comparing the efficienc of the nodes with respect to receiving and sending information being indicative of the rapidity with which this transport happens. The objective of this work is to study the process of random walk and use it to analyze the efficiency of Centrality measures, inferring the number of random walkers who pass by us in complex networks. Measures of centrality are analyzed centralities Degree, Centrality Intermediation by Minor Roads, Centralization of Random Walk. To compare the efficiency of these measures of centrality in the different network models, numerical simulations were performed. With these, it was noticed that the behavior of the diffusion of walkers varies for each network model. Random network for the flow of walkers from the evenly is not possible to highlight some vertex of utmost importance within the network. It can be observed that the measure of centrality of Random Walk is the one that showed greater efficiency by pointing a greater flow of walkers to the vertices that had a higher value for this measure. / Os estudos sobre redes complexas nos auxiliam a compreender e modelar muitas situações do mundo real. O mundo é abundante em redes que podem ser encontradas em diversos contextos reais. O termo redes faz referência às relações estabelecidas entre dois conjuntos e podem ser representadas por meio da teoria de grafos. A classificação das redes complexas se dá de acordo com os modelos criados para representá-las, tais como as redes Aleatórias, redes de Pequeno Mundo, redes Sem Escala e redes Hierárquicas. Dentro da perspectiva de redes complexas, um estudo que pode trazer contribuições importantes é análise do fenômeno de difusão de informação em redes, os quais podem ser entendidos através do processo da caminhada aleatória, a qual se caracteriza por ser um processo estocástico utilizado como um mecanismo de transporte e pesquisa em redes complexas. A caminhada aleatória nas redes complexas pode ser utilizada para verificar o comportamento de cada modelo de rede frente à dissipação. Cada modelo de rede apresenta um comportamento diferente com relação ao número de caminhantes aleatórios que passam por nó da rede ao longo do tempo. Este número de caminhantes irá depender da estrutura das redes geradas por cada modelo e das medidas de Centralidade de cada nó. As medidas de centralidade dos vértices da rede são úteis para comparar a eficiência dos nós com relação ao recebimento e envio de informações sendo indicativos da rapidez com a qual, este transporte acontece. O objetivo deste trabalho é estudar o processo da caminhada aleatória e utilizá-la para analisar a eficiência das medidas de Centralidade, inferindo o número de caminhantes aleatórios que passam pelos nós nas redes complexas. As medidas de centralidade analisadas são as centralidades do Grau, Centralidade de Intermediação por Menores Caminhos, Centralidade da Caminhada Aleatória. Para comparar a eficiência das referidas medidas de Centralidade nos diferentes modelos de redes, foram realizadas simulações numéricas. Com estas, percebeu-se que o comportamento da difusão de caminhantes varia para cada modelo de rede. Para a rede Aleatória o fluxo de caminhantes se da de maneira uniforme não sendo possível destacar algum vértice de maior importância dentro da rede. Pode-se observar que a medida de Centralidade da Caminhada Aleatória é a que mostrou maior eficiência ao apontar o um maior fluxo de caminhantes aos vértices que possuíam um maior valor para essa medida.

O passeio aleatório quântico foi totalmente entendido por [3] e desde então muitos esforços foram feitos para compreender casos mais gerais como no passeio aleatório tradicional. Nós introduzimos o caso periódico e discutimos a heurística sendo considerada como uma partícula quântica difundindo em um cristal atômico linear. Assim, estendemos o teorema de Grimmett-Janson- Scudo [3] para este caso que é um método para obter a densidade de probabilidade limite do operador posição dependendo da diagonalização da matriz de evolução unitária e mostramos que o caso periódico é de fato balístico, [9]. Como um exemplo, édiscutida a densidade probabilidade limite de período dois. / The homogeneous quantum random walk was completely understood by [3] and since then many efforts were made to compreehend more general cases like in the tradicional random walk. We introduce the periodic case and discuss a heuristic to be considered as a quantum particle diffusion in a atomic linear crystal. Thus, we extend the theorem of Grimmett-Janson-Scudo [3] to this case which is a method to obtain the limit of the probability density of the position operator depending on the diagonalization of the unitary evolution matrix and show that the periodic case is in fact ballistic, [9]. As an example, it is shown the limit probability density of the period two.

Declines in amphibian populations have prompted study of their ecology and distribution. The purpose of this study was to survey two sites located within different physiographic and one herpetofaunal region of Northeast Tennessee, comparing species composition and activity. The first, Henderson Wetland, is in the Appalachian Ridge and Valley physiographic region. The second, John's Bog, is in the Blue Ridge. Survey methods included random walks, aural surveys, and point source collections during a 16-month period (February 1999 to May 2000). Nine caudate (Plethodontidae) and one anuran species (Ranidae) were found in John's Bog. Seven caudate (Ambystomatidae, Plethodontidae, Salamandridae) and five anuran species (Hylidae, Ranidae) were found in Henderson Wetland. Assemblages were compared using an index of community similarity. Sites differed regarding amphibians detected. Temporal activity was not compared because of different species compositions. Instead, temporal data were compared to literature. Data will be used in future amphibian studies and site management.

We establish a strong connection between buildings and Hecke algebras through the study of two algebras of averaging operators on buildings. To each locally finite regular building we associate a natural algebra B of chamber set averaging operators, and when the building is affine we also define an algebra A of vertex set averaging operators. In the affine case, it is shown how the building gives rise to a combinatorial and geometric description of the Macdonald spherical functions, and of the centers of affine Hecke algebras. The algebra homomorphisms from A into the complex numbers are studied, and some associated spherical harmonic analysis is conducted. This generalises known results concerning spherical functions on groups of p-adic type. As an application of this spherical harmonic analysis we prove a local limit theorem for radial random walks on affine buildings.

In this thesis we develop and use a continuum random walk framework to solve problems that are usually studied using a discrete random walk on a discrete lattice. Problems studied include; the time it takes for a random walker to be absorbed at a trap on a fractal lattice, the calculation of the spectral dimension for several different classes of networks, the calculation of the density of states for a multi-layered Bethe lattice and the relationship between diffusion exponents and a resistivity exponent that occur in relevant power laws. The majority of the results are obtained by deriving an expression for a Laplace transformed Green’s function or first passage time, and then using Tauberian theorems to find the relevant asymptotic behaviour. The continuum framework is established by studying the diffusion equation on a 1-d bar with non-homogeneous boundary conditions. The result is extended to model diffusion on networks through linear algebra. We derive the transformation linking the Green’s functions and first passage time results in the continuum and discrete settings. The continuum method is used in conjunction with renormalization techniques to calculate the time taken for a random walker to be absorbed at a trap on a fractal lattice and also to find the spectral dimension of new classes of networks. Although these networks can be embedded in the d- dimensional Euclidean plane, they do not have a spectral dimension equal to twice the ratio of the fractal dimension and the random walk dimension when the random walk on the network is transient. The networks therefore violate the Alexander-Orbach law. The fractal Einstein relationship (a relationship relating a diffusion exponent to a resistivity exponent) also does not hold on these networks. Through a suitable scaling argument, we derive a generalised fractal Einstein relationship which holds for our lattices and explains anomalous results concerning transport on diffusion limited aggregates and Eden trees.