Graph libraries containing already-implemented algorithms are highly desired since users can conveniently use the algorithms off-the-shelf to achieve fast analyt- ics and prototyping, rather than implementing the algorithms with lower-level APIs. Besides the ease of use, the ability to efficiently process extra large graphs is also required by users. The popular existing graph libraries include the igraph R library and the NetworkX Python library. Although these libraries provide many off-the-shelf algorithms for users, the in-memory graph representation limits their scalability for computing on large graphs. Therefore, in this work, we develop Graph-XLL: a graph library implemented using the WebGraph framework in a vertex-centric manner, with much less memory requirement compared to igraph and NetworkX. Scalable analytics for extra large graphs (up to tens of millions of vertices and billions of edges) can be achieved on a single consumer grade machine within a reasonable amount of time. Such computation would cause out-of-memory error if using igraph or NetworkX. / Graduate

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Thesis (Ph. D.)--West Virginia University, 2009. / Title from document title page. Document formatted into pages; contains vi, 84 p. : ill. Includes abstract. Includes bibliographical references (p. 71-74).

Thesis (Ph. D.)--West Virginia University, 2005. / Title from document title page. Document formatted into pages; contains vi, 49 p. : ill. Includes abstract. Includes bibliographical references (p. 47-49).

Thesis (M.S.)--The University of North Carolina at Greensboro, 2008. / Directed by Paul Duvall; submitted to the Dept. of Mathematical Sciences. Title from PDF t.p. (viewed Sep. 4, 2009). Includes bibliographical references (p. 57).

Thesis (M.S.)--West Virginia University, 2005. / Title from document title page. Document formatted into pages; contains v, 17 p. : ill. Includes abstract. Includes bibliographical references (p. 17).

Ph.D. / In this thesis, our primary objective is to investigate the effects that various graph modifications have on the total domination number of a graph. In Chapter 1, we introduce basic graph theory concepts and preliminary definitions. In Chapters 2 and 3, we investigate the graph modification of edge removal. In Chapter 2, we characterize graphs for which the removal of any arbitrary edge increases the total domination number. We also begin the investigation of graphs for which the removal of any arbitrary edge has no effect on the total domination number. In Chapter 3, we continue this investigation and determine the minimum number of edges required for these graphs. The contents of Chapters 2 and 3 have been published in Discrete Applied Mathematics [15] and [16]. In Chapter 4, we investigate the graph modification of edge addition. In particular, we focus our attention on graphs for which adding an edge between any pair of nonadjacent vertices has no effect on the total domination number. We characterize these graphs, determine a sharp upper bound on their total domination number and determine which combinations of order and total domination number are attainable. 10 11 We also study claw-free graphs which have this property. The contents of this chapter were published in Discrete Mathematics [20]. In Chapter 5, we investigate the graph modification of vertex removal. We characterize graphs for which the removal of any vertex changes the total domination number and find sharp upper and lower bounds on the total domination number of these graphs. We also characterize graphs for which the removal of an arbitrary vertex has no effect on the total domination number and we further show that they have no forbidden subgraphs. The contents of this chapter were published in Discrete Applied Mathematics [14]. In Chapters 6 and 7, we investigate the graph modification of edge lifting. In Chapter 6, we show that there are no trees for which every possible edge lift decreases the domination number, and we characterize trees for which every possible edge lift increases the domination number. The contents of Chapter 6 were published in the journal Quaestiones Mathematicae [17]. In Chapter 7, we show that there are no trees for which every possible edge lift decreases the total domination number and that there are no trees for which every possible edge lift leaves the total domination number unchanged. We characterize trees for which every possible edge lift increases the total domination number. At the time of the writing of this thesis, the contents of Chapter 7 have been published online in the Journal of Combinatorial Optimization [18] and will appear in print in a future issue.

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Large-scale graph analysis and visualization is becoming a more challenging task, due to the increasing amount of graph data. This dissertation focuses on methods to ease the task of exploring large-scale graphs through graph sampling and visualization. Graph sampling aims to reduce the complexity of graph drawing, while preserving properties of the original graph, allowing analysis of the smaller sample which yields the characteristics similar to those of the original graph. Graph visualization is an effective and intuitive approach to observing structures within graph data. For large-scale graphs, graph sampling and visualization are straightforward strategies to gain insights into common issues that are often encountered. This dissertation evaluates commonly used graph sampling methods through a combined visual and statistical comparison of graphs sampled at various rates based on random graphs, small-world graphs, scaleree graphs, and real-world graphs. This benchmark study can be used as a guideline in choosing the appropriate method for a particular graph sampling task. In addition, this thesis proposes three types of distributed sampling algorithms and develops a sampling package on Spark. Compared with traditional/non-distributed graph sampling approaches, the scalable distributed sampling approaches are as reliable as the traditional/non-distributed graph sampling techniques, and they bring much needed improvement to sampling efficiency, especially with regards to topology-based sampling. This benchmark study in traditional/non-distributed graph sampling is also applicable to distributed graph sampling as well. A contribution to the area of graph visualization is also made through the presentation of a scalable graph visualization system-BGS (Big Graph Surfer) that creates hierarchical structure from an original graph and provides interactive navigation along the hierarchy by expanding or collapsing clusters when visualizing large-scale graphs. A distributed computing framework-Spark provides the backend for BGS on clustering and visualization. This architecture makes it capable of visualizing a graph up to 1 billion nodes or edges in real-time. In addition, BGS provides a series of hierarchy and graph exploration methods, such as hierarchy view, hierarchy navigation, hierarchy search, graph view, graph navigation, graph search, and other useful interactions. These functionalities facilitate the exploration of very large-scale graphs. Evaluation of BGS is performed through application to several representative of large-scale graph datasets and comparison with other existing graph visualization tools in scalability, usability, and flexibility. The dissertation concludes with a summarization of the contributions and their improvement on large-scale graph analysis and visualization, and a discussion about possible future work on this research field.

Graph databases offer an efficient way to store and access inter-connected data. However, to query large graphs that no longer fit in memory, it becomes necessary to make multiple trips to the storage device to filter and gather data based on the query. But I/O accesses are expensive operations and immensely slow down query response time and prevent us from fully exploiting the graph specific benefits that graph databases offer. The storage models of most existing graph database systems view graphs as indivisible structures and hence do not allow a hierarchical layering of the graph. This adversely affects query performance for large graphs as there is no way to filter the graph on a higher level without actually accessing the entire information from the disk. Distributing the storage and processing is one way to extract better performance. But current distributed solutions to this problem are not entirely effective, again due to the indivisible representation of graphs adopted in the storage format. This causes unnecessary latency due to increased inter-processor communication. In this dissertation, we propose an optimized distributed graph storage system for scalable and faster querying of big graph data. We start with our unique physical storage model, in which the graph is decomposed into three different levels of abstraction, each with a different storage hierarchy. We use a hybrid storage model to store the most critical component and restrict the I/O trips to only when absolutely necessary. This lets us actively make use of multi-level filters while querying, without the need of comprehensive indexes. Our results show that our system outperforms established graph databases for several class of queries. We show that this separation also eases the difficulties in distributing graph data and go on propose a more efficient distributed model for querying general purpose graph data using the Spark framework.