This dissertation explores two themes central to the field of non-equilibrium statistical physics. The first is centered around the use of random walks, first-passage processes, and Brownian motion to model basic stochastic search processes found in biology and ecological systems. The second is centered around clustered networks: how clustering modifies the nature of transition in the appearance of various graph motifs and their use in modeling social networks. In the first part of this dissertation, we start by investigating properties of intermediate crossings of Brownian paths. We develop simple analytical tools to obtain probability distributions of intermediate crossing positions and intermediate crossing times of Brownian paths. We find that the distribution of intermediate crossing times can be unimodal or bimodal. Next, we develop analytical and numerical methods to solve a system of 𝑁 diffusive searchers which are reset to the origin at stochastic or periodic intervals. We obtain the optimal criteria to search for a fixed target in one, two and three dimensions. For these two systems, we also develop efficient ways to simulate Brownian paths, where the simulation kernel makes maximal use of first-passage ideas. Finally we develop a model to understand foraging in a resource-rich environment. Specifically, we investigate the role of greed on the lifetime of a diffusive forager. This lifetime shows non-monotonic dependence on greed in one and two dimensions, and surprisingly, a peak for negative greed in 1d. In the second part of this dissertation, we develop simple models to capture the non-tree-like (clustering) aspects of random networks that arise in the real world. By 'clustered networks', we specifically mean networks where the probability of links between neighbors of a node (i.e., 'friends of friends') is positive. We discuss three simple and related models. We find a series of transitions in the density of graph motifs such as triangles (3-cliques), 4-cliques etc as a function of the clustering probability. We also find that giant 3-cores emerge through first- or second-order, or even mixed transitions in clustered networks.

vi, 48 p. : ill. A print copy of this thesis is available through the UO Libraries. Search the library catalog for the location and call number. / We compare the relaxation times of two random walks - the simple random walk and the metropolis walk - on an arbitrary finite multigraph G. We apply this result to the random graph with n vertices, where each edge is included with probability p = [Special characters omitted.] where λ > 1 is a constant and also to the Newman-Watts small world model. We give a bound for the reconstruction problem for general trees and general 2 × 2 matrices in terms of the branching number of the tree and some function of the matrix. Specifically, if the transition probabilities between the two states in the state space are a and b , we show that we do not have reconstruction if Br( T ) [straight theta] < 1, where [Special characters omitted.] and Br( T ) is the branching number of the tree in question. This bound agrees with a result obtained by Martin for regular trees and is obtained by more elementary methods. We prove an inequality closely related to this problem. / Committee in charge: David Levin, Chairperson, Mathematics; Christopher Sinclair, Member, Mathematics; Marcin Bownik, Member, Mathematics; Hao Wang, Member, Mathematics; Van Kolpin, Outside Member, Economics

We study a family of random walks defined on certain Euclidean lattices that are related to incidence matrices of balanced incomplete block designs. We estimate the return probability of these random walks and use it to determine the asymptotics of the number of balanced incomplete block design matrices. We also consider the problem of collisions of independent simple random walks on graphs. We prove some new results in the collision problem, improve some existing ones, and provide counterexamples to illustrate the complexity of the problem.

The diploma thesis deals with the questions considering field-walks in the surrounding of the town Zliv. There are six walks suggested {--} Wandering in a forest, Around a lake in quiet, Life of the animals in a forest. At the lake `` Mydlák{\crqq}, Zliv and We get to know our meadows. The walks are designed for pupils in the fourth and fifth grade at the primary schools; each walk encloses a work sheet. Their target is to deepen and fix pupils´knowledge and understanding.

O avanço das pesquisas em redes complexas proporcionou desenvolvimentos significativos para a compreensão de sistemas complexos. Uma rede complexa é modelada matematicamente por meio de um grafo, onde cada vértice representa uma unidade dinâmica e suas interações são simbolizadas por um conjunto de arestas. Para se determinar propriedades estruturais desse sistema, caminhadas aleatórias tem-se mostrado muito úteis pois dependem apenas de informações locais (vértices vizinhos). Entre elas, destaca-se o passeio auto-excludente (SAW) que possui a restrição de não visitar um vértice que já foi alcançado, ou seja, apresenta memória do caminho percorrido. Por este motivo o SAW tem apresentado melhores resultados do que caminhantes sem restrição, na exploração da rede. Entretanto, por não se tratar de um processo Markoviano ele apresenta grande complexidade analítica, tornando indispensável o uso de simulações computacionais para melhor compreensão de sua dinâmica em diferentes topologias. Mesmo com as dificuldades analíticas, o SAW se tornou uma ferramenta promissora na identificação de estruturas de comunidades. Apesar de sua importância, detecção de comunidades permanece um problema em aberto devido à alta complexidade computacional associada ao problema de optimização, além da falta de uma definição formal do significado de comunidade. Neste trabalho, propomos um método de detecção de comunidades baseado em SAW para extrair uma estrutura de comunidades da rede otimizando o parâmetro modularidade. Combinamos características extraídas desta dinâmica com a análise de componentes principais para posteriormente classificar os vértices em grupos por meio da clusterização hierárquica aglomerativa. Para avaliar a performance deste novo algoritmo, comparamos os resultados com outras quatro técnicas populares: Girvan-Newman, Fastgreedy, Walktrap e Infomap, aplicados em dois tipos de redes sintéticas e nove redes reais diversificadas e bem conhecidas. Para os benchmarks, esta nova técnica produziu resultados satisfatórios em diferentes combinações de parâmetros, como tamanho de rede, distribuição de grau e número de comunidades. Já para as redes reais, obtivemos valores de modularidade superior aos métodos tradicionais, indicando uma distribuição de grupos mais adequada à realidade. Feito isso, generalizamos o algoritmo para redes ponderadas e digrafos, além de incorporar metadados à estrutura topológica a fim de melhorar a classificação em grupos. / The progress in complex networks research has provided significant understanding of complex systems. A complex network is mathematically modeled by a graph, where each vertex represents a dynamic unit and its interactions are symbolized by groups of edges. To determine the system structural properties, random walks have shown to be a useful tool since they depend only on local information (neighboring vertices). Among them, the selfavoiding walk (SAW) stands out for not visiting vertices that have already been reached, meaning it can record the path that has been travelled. For this reason, SAW has shown better results when compared to non-restricted walkers network exploration methods. However, as SAW is not a Markovian process, it has a great analytical complexity and needs computational simulations to improve its dynamics in different topologies. Even with the analytical complexity, SAW has become a promising tool to identify the community structure. Despite its significance, detecting communities remains an unsolved problem due to its high computational complexity associated to optimization issues and the lack of a formal definition of communities. In this work, we propose a method to identify communities based on SAW to extract community structure of a network through optimization of the modularity score. Combining technical features of this dynamic with principal components analyses, we classify the vertices in groups by using hierarchical agglomerative clustering. To evaluate the performance of this new algorithm, we compare the results with four other popular techniques: Girvan-Newman, Fastgreedy, Walktrap and Infomap, applying the algorithm in two types of synthetic networks and nine different and well known real ones. For the benchmarks, this new technique shows satisfactory results for different combination of parameters as network size, degree distribution and number of communities. As for real networks, our data shows better modularity values when compared to traditional methods, indicating a group distribution most suitable to reality. Furthermore, the algorithm was adapted for general weighted networks and digraphs in addition to metadata incorporated to topological structure, in order to improve the results of groups classifications.

Doctor of Philosophy / Department of Electrical and Computer Engineering / Pietro Poggi-Corradini / Caterina M. Scoglio / The modulus of a family of walks quantifies the richness of the family by favoring having many short walks over a few longer ones. In this dissertation, we investigate various families of walks to study new measures for quantifying network properties using modulus. The proposed new measures are compared to other known quantities. Our proposed method is based on walks on a network, and therefore will work in great generality. For instance, the networks we consider can be directed, multi-edged, weighted, and even contain disconnected parts. We study the popular centrality measure known in some circles as information centrality, also known as effective conductance centrality. After reinterpreting this measure in terms of modulus of families of walks, we introduce a modification called shell modulus centrality, that relies on the egocentric structure of the graph. Ego networks are networks formed around egos with a specific order of neighborhoods. We then propose efficient analytical and approximate methods for computing these measures on both directed and undirected networks. Finally, we describe a simple method inspired by shell modulus centrality, called general degree, which improves simple degree centrality and could prove to be a useful tool for practitioners in the applied sciences. General degree is useful for detecting the best set of nodes for immunization. We also study the structure of loops in networks using the notion of modulus of loop families. We introduce a new measure of network clustering by quantifying the richness of families of (simple) loops. Modulus tries to minimize the expected overlap among loops by spreading the expected link-usage optimally. We propose weighting networks using these expected link-usages to improve classical community detection algorithms. We show that the proposed method enhances the performance of certain algorithms, such as spectral partitioning and modularity maximization heuristics, on standard benchmarks. Computing loop modulus benefits from efficient algorithms for finding shortest loops, thus we propose a deterministic combinatorial algorithm that finds a shortest cycle in graphs. The proposed algorithm reduces the worst case time complexity of the existing combinatorial algorithms while visiting at most the cycle basis. For most empirical networks with sublinear average degree our algorithm is subcubic.

We consider transformations of a given random walk on a countable group determined by Markov stopping times. We prove that these transformations preserve the Poisson boundary. Moreover, under some mild conditions, the asymptotic entropy (resp., rate of escape) of the transformed random walks is equal to the asymptotic entropy (resp., rate of escape) of the original random walk multiplied by the expectation of the corresponding stopping time. This is an analogue of the well-known Abramov's formula from ergodic theory.

In this thesis, we look into the problem of forming and maintaining good committees that can represent a distributed network. The solution to this problem can be used as a sub-routine for Byzantine Agreement that only costs sub-quadratic message complexity. Most importantly, we make no cryptographic assumptions such as the Random Oracle assumption and the existence of private channels. However, we do assume the network to be peer-to-peer, where a message receiver knows who the message sender is. Under the synchronous full information model, our solution is to utilize an approximating disperser for selecting a good next committee with high probability, repeatedly. We consider several existing theoretical constructions (randomized and deterministic) for approximating dispersers, and examine the practical applicability of them, while improving constants for some constructions. This algorithm is robust against a semi-adaptive adversary who can decide the set of nodes to corrupt periodically. Thus, a new committee should be selected before the current committee gets corrupted. We also prove some constructions that do not work practically for our scenario. / Graduate

Indiana University-Purdue University Indianapolis (IUPUI) / Over the last two decades a new theory has been developed and intensively investigated in quantum physics. The theory stipulates that a non-Hermitian Hamiltonian can also represents a physical system as long as its energy spectra can be purely real in certain regime depending on the parameters of the Hamiltonian. It was demonstrated that the reality of the eigenenergy was conditioned by a certain kind of symmetry embedded in the actual non-Hermitian system. Indeed, such systems have a combined reflection (parity) symmetry (P) and time-reversal symmetry (T), PT-symmetry. The theory opens the door to new features particularly in open systems in which there could be gain and/or loss of particle or energy from and/or to the environment. A key property of the theory is the PT-symmetry breaking transition which occurs at the exceptional point (EP). The exceptional points are special degeneracies characterized by a coalescence of not only the eigenvalues but also of the corresponding eigenvectors of the system; and the coalescence happens when the gain-loss strength, a measure of the openness of the system, exceeds the intrinsic energy-scale of the system. In recent years, quantum walks with PT-symmetric non-unitary time evolution have been realized in systems with balanced gain and loss. These systems fall in two categories namely continuous time quantum walks (CTQW) that are characterized by a unitary or non-unitary time evolution Hamiltonian, and discrete-time quantum walks (DTQW) whose dynamic is described by a unitary or non-unitary time evolution operator consisting of a product of shift, coin, and gain-loss operations. In this thesis, we investigate the PT-symmetric phase of CTQW and DTQW in a variety of non-Hermitian lattice systems with both position-dependent and position independent, parity-symmetric tunneling functions in the presence of PT-symmetric impurities located at arbitrary parity-symmetric site on the lattice. Moreover, we explore the topological phase diagram and its novel features in non-Hermitian, homogeneous and non-homogeneous, PT-symmetric DTQW with closed and open boundary conditions. We conduct our study using analytical and numerical approaches that are directly and easily implementable in physical experiments. Among others, we found that, despite their non-unitary evolution, open systems governed by parity-time symmetric Hamiltonian support conserved quantities and that the PT-symmetry breaking threshold depends on the physical structure of the Hamiltonian and its underlying symmetries.

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