We study the large deviation behaviour of simple random walks in dimension three or more in this thesis. The first part of the thesis concerns the number of lattice sites visited by the random walk. We call this the range of the random walk. We derive a large deviation principle for the probability that the range of simple random walk deviates from its mean. Our result describes the behaviour for deviation below the typical value. This is a result analogous to that obtained by van den Berg, Bolthausen, and den Hollander for the volume of the Wiener sausage. In the second part of the thesis, we are interested in the number of lattice sites visited by two independent simple random walks starting at the origin. We call this the intersection of ranges. We derive a large deviation principle for the probability that the intersection of ranges by time n exceeds a multiple of n. This is also an analogous result of the intersection volume of two independent Wiener sausages.

This paper investigates whether the inherent non-stationarity of macroeconomic time series is entirely due to a random walk or also to non-linear components. Applying the numerical tools of the analysis of dynamical systems to long time series for the US, we reject the hypothesis that these series are generated solely by a linear stochastic process. Contrary to the Real Business Cycle theory that attributes the irregular behavior of the system to exogenous random factors, we maintain that the fluctuations in the time series we examined cannot be explained only by means of external shocks plugged into linear autoregressive models. A dynamical and non-linear explanation may be useful for the double aim of describing and forecasting more accurately the evolution of the system. Linear growth models that find empirical verification on linear econometric analysis, are therefore seriously called in question. Conversely non-linear dynamical models may enable us to achieve a more complete information about economic phenomena from the same data sets used in the empirical analysis which are in support of Real Business Cycle Theory. We conclude that Real Business Cycle theory and more in general the unit root autoregressive models are an inadequate device for a satisfactory understanding of economic time series. A theoretical approach grounded on non-linear metric methods, may however allow to identify non-linear structures that endogenously generate fluctuations in macroeconomic time series. (authors' abstract) / Series: Working Papers Series "Growth and Employment in Europe: Sustainability and Competitiveness"

Existing methods to utilize capacity-heterogeneity in a P2P system either rely on constructing special overlays with capacity-proportional node degree or use topology adaptation to match a node's capacity with that of its neighbors. In existing P2P networks, which are often characterized by diverse node capacities and high churn, these methods may require large node degree or continuous topology adaptation, potentially making them infeasible due to their high overhead. In this thesis, we propose an unstructured P2P system that attempts to address these issues. We first prove that the overall throughput of search queries in a heterogeneous network is maximized if and only if traffic load through each node is proportional to its capacity. Our proposed system achieves this traffic distribution by biasing search walks using the Metropolis-Hastings algorithm, without requiring any special underlying topology. We then define two saturation metrics for measuring the performance of overlay networks: one for quantifying their ability to support random walks and the second for measuring their potential to handle the overhead caused by churn. Using simulations, we finally compare our proposed method with Gia, an existing system which uses topology adaptation, and find that the former performs better under all studied conditions, both saturation metrics, and such end-to-end parameters as query success rate, latency, and query-hits for various file replication schemes.

Online Social Networks (OSNs) such as Facebook, Twitter, and YouTube are among the most popular sites on the Internet. Billions of users are connected through these sites, building strong and effective communities to share views and ideas, and make recommendations nowadays. Therefore, by choosing an appropriate user-base from billions of people is required to analyze the structure and key characteristics of the large social graphs to improve current systems and to design new applications. For this reason, node sampling technique plays an important role to study large-scale social networks. As a basic requirement, the sampled nodes and their links should possess similar statistical features of the original network, otherwise the conclusion drawn from the sampled network may not be appropriate for the entire population. Hence, good sampling strategies are key to many online social network applications. For instance, before introducing a new product or adding new feature(s) of a product to the online social network community, that specific new product or the additional feature has to be exposed to only a small set of users, who are carefully chosen to represent the complete set of users. As such, different random walk-based sampling techniques have been introduced to produce samples of nodes that not only are internally well-connected but also capture the statistical features of the whole network. Traditionally, walk-based techniques do not have the restriction on the number of times that a node can be re-visited while sampling. This may lead to an inefficient sampling method, because the walk may be "stuck" at a small number of high-degree nodes without being able to reach out to the rest of the nodes. A random walk, even after a large number of hops, may not be able to obtain a sampled network that captures the statistical features of the entire network. In this thesis, we propose two walk-based sampling techniques to address the above problem, called K-Avoiding Random Walk (KARW) and Neighborhood-Avoiding Random Walk (NARW). With KARW, the number of times that a node can be re-visited is constrained within a given number K. With NARW, the random walk works in a "jump" fashion, since the walk starts outside of the N-hop neighborhood from the current node chosen randomly. By avoiding the current nodes neighboring area of level-N, NARW is expected to reach out the other nodes within the entire network quickly. We apply these techniques to construct multiple independent subgraphs from a social graph, consisting of 63K users with around a million connections between users collected from a Facebook dataset. By simulating our proposed strategies, we collect performance metrics and compare the results with the current state-of-the-art sampling techniques (Uniform Random Sampling, Random Walk, and Metropolis Hastings Random Walk). We also calculate some of the key statistical features (i.e., degree distribution, betweenness centrality, closeness centrality, modularity, and clustering coefficient) of the sampled graphs to get an idea about the network structures that essentially represent the original social graph. / Graduate / 0984 / shahed.anwar@gmail.com

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Thesis (M.Sc.)--University of Adelaide, Dept. of Applied Mathematics, 2001. / "November 2001." Bibliography: leaves 128-130.

Thesis (Ph. D.)--University of Washington, 2002. / Vita. Includes bibliographical references (leaves 106-114).

Thesis (Ph. D.)--Rutgers University, 2007. / "Graduate Program in Computer Science." Includes bibliographical references (p. 94-100).