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There is a long history of research in theoretical computer science devoted to designing efficient algorithms for graph problems. In many modern applications the graph in question is changing over time, and we would like to avoid rerunning our algorithm on the entire graph every time a small change occurs. The evolving nature of graphs motivates the dynamic graph model, in which the goal is to minimize the amount of work needed to reoptimize the solution when the graph changes. There is a large body of literature on dynamic algorithms for basic problems that arise in graphs. This thesis presents several improved dynamic algorithms for two fundamental graph problems: shortest paths, and matching.

Fung, Yu Hin. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2011. / Includes bibliographical references (leaves 39-40). / Abstracts in English and Chinese. / List of Figures --- p.v / List of Tables --- p.vii / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- Extremogram --- p.3 / Chapter 2.1 --- Strictly Stationary --- p.3 / Chapter 2.2 --- Regularly Varying: A time series {Xt} --- p.3 / Chapter 2.3 --- (Upper) tail dependence --- p.5 / Chapter 2.4 --- Extremogram --- p.6 / Chapter 3 --- Simulated Models --- p.9 / Chapter 3.1 --- Autoregressive (AR) Process --- p.9 / Chapter 3.1.1 --- The simulation --- p.9 / Chapter 3.1.2 --- Theoretical findings --- p.11 / Chapter 3.2 --- Moving Average (MA) Process --- p.12 / Chapter 3.2.1 --- The simulation --- p.12 / Chapter 3.3 --- GARCH and SV --- p.25 / Chapter 4 --- Applications to Market Data --- p.29 / Chapter 4.1 --- Case study: 2011 Japan Earthquake EOD data --- p.29 / Chapter 4.1.1 --- Data description --- p.29 / Chapter 4.1.2 --- Results --- p.30 / Chapter 4.2 --- Case study: TEPCO multi-timeframe analysis --- p.31 / Chapter 4.2.1 --- Data description --- p.31 / Chapter 4.2.2 --- Results --- p.32 / Chapter 5 --- Summary --- p.37 / References --- p.39

This thesis explains to statisticians what graphical models are and how to use them for statistical inference; in particular, how to use decomposable graphical models for efficient inference in covariance selection and multivariate regression problems. The first aim of the thesis is to show that decomposable graphical models are worth using within a Bayesian framework. The second aim is to make the techniques of graphical models fully accessible to statisticians. To achieve these aims the thesis makes a number of statistical contributions. First, it proposes a new prior for decomposable graphs and a simulation methodology for estimating this prior. Second, it proposes a number of Markov chain Monte Carlo sampling schemes based on graphical techniques. The thesis also presents some new graphical results, and some existing results are reproved to make them more readily understood. Appendix 8.1 contains all the programs written to carry out the inference discussed in the thesis, together with both a summary of the theory on which they are based and a line by line description of how each routine works.

Thesis (M.S.)--West Virginia University, 2002. / Title from document title page. Document formatted into pages; contains vi, 96 p. : ill. Includes abstract. Includes bibliographical references (p. 91-96).

With the increasing popularity of social networking applications, large volumes of graph data are becoming available. Large graphs are also derived by structure extraction from relational, text, or scientific data (e.g., relational tuple networks, citation graphs, ontology networks, protein-protein interaction graphs). Nodeto-node proximity is the key building block for many graph based applications that search or analyze the data. Among various proximity measures, random walk with restart (RWR) is widely adapted because of its ability to consider the global structure of the whole network. Although RWR-based similarity search has been well studied before, there is no prior work on reverse top-k proximity search in graphs based on RWR. We discuss the applicability of this query and show that the direct application of existing methods on RWR-based similarity search to solve reverse top-k queries has very high computational and storage demands. To address this issue, we propose an indexing technique, paired with an on-line reverse top-k search algorithm. In the indexing step, we compute from the graph G a graph index, which is based on a K X |V| matrix, containing in each column v the K largest approximate proximity values from v to any other node in G. K is application-dependent and represents the highest value of k in a practical reverse top-k query. At each column v of the index, the approximate values are lower bounds of the K largest proximity values from v to all other nodes. Given the graph index and a reverse top-k query q (k _ K), we prove that the exact proximities from any node v to query q can be efficiently computed by applying the power method. By comparing these with the corresponding lower bounds taken from the k-th row of the graph index, we are able to determine which nodes are certainly not in the reverse top-k result of q. For some of the remaining nodes, we may also be able to determine that they are certainly in the reverse top-k result of q, based on derived upper bounds for the k-th largest proximity value from them. Finally, for any candidate that remains, we progressively refine its approximate proximities, until based on its lower or upper bound it can be determined not to be or to be in the result. The proximities refined during a reverse top-k are used to update the graph index, making its values progressively more accurate for future queries. Our experimental evaluation shows that our technique is efficient and has manageable storage requirements even when applied on very large graphs. We also show the effectiveness of the reverse top-k search in the scenarios of spam detection and determining the popularity of authors. / published_or_final_version / Computer Science / Master / Master of Philosophy

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