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.

The study and recognition of graph families (or graph properties) is an essential part of combinatorics. Graph colouring is another fundamental concept of graph theory that can be looked at, in large part, as the recognition of a family of graphs that are colourable according to certain rules. In this thesis, we study additive induced-hereditary families, and some generalisations, from a colouring perspective. Our main results are: · Additive induced-hereditary families are uniquely factorisable into irreducible families. · If <i>P</i> and <i>Q</i> are additive induced-hereditary graph families, then (<i>P</i>,<i>Q</i>)-COLOURING is NP-hard, with the exception of GRAPH 2-COLOURING. Moreover, with the same exception, (<i>P</i>,<i>Q</i>)-COLOURING is NP-complete iff <i>P</i>- and <i>Q</i>-RECOGNITION are both in NP. This proves a 1997 conjecture of Kratochvíl and Schiermeyer. We also provide generalisations to somewhat larger families. Other results that we prove include: · a characterisation of the minimal forbidden subgraphs of a hereditary property in terms of its minimal forbidden induced-subgraphs, and <i>vice versa</i>; · extensions of Mihók's construction of uniquely colourable graphs, and Scheinerman's characterisations of compositivity, to disjoint compositive properties; · an induced-hereditary property has at least two factorisations into arbitrary irreducible properties, with an explicitly described set of exceptions; · if <i>G</i> is a generating set for <i>A</i> &#959; <i>B</i>, where <i>A</i> and <i>B</i> are indiscompositive, then we can extract generating sets for <i>A</i> and <i>B</i> using a <i>greedy algorithm</i>.

The study and recognition of graph families (or graph properties) is an essential part of combinatorics. Graph colouring is another fundamental concept of graph theory that can be looked at, in large part, as the recognition of a family of graphs that are colourable according to certain rules. In this thesis, we study additive induced-hereditary families, and some generalisations, from a colouring perspective. Our main results are: · Additive induced-hereditary families are uniquely factorisable into irreducible families. · If <i>P</i> and <i>Q</i> are additive induced-hereditary graph families, then (<i>P</i>,<i>Q</i>)-COLOURING is NP-hard, with the exception of GRAPH 2-COLOURING. Moreover, with the same exception, (<i>P</i>,<i>Q</i>)-COLOURING is NP-complete iff <i>P</i>- and <i>Q</i>-RECOGNITION are both in NP. This proves a 1997 conjecture of Kratochvíl and Schiermeyer. We also provide generalisations to somewhat larger families. Other results that we prove include: · a characterisation of the minimal forbidden subgraphs of a hereditary property in terms of its minimal forbidden induced-subgraphs, and <i>vice versa</i>; · extensions of Mihók's construction of uniquely colourable graphs, and Scheinerman's characterisations of compositivity, to disjoint compositive properties; · an induced-hereditary property has at least two factorisations into arbitrary irreducible properties, with an explicitly described set of exceptions; · if <i>G</i> is a generating set for <i>A</i> &#959; <i>B</i>, where <i>A</i> and <i>B</i> are indiscompositive, then we can extract generating sets for <i>A</i> and <i>B</i> using a <i>greedy algorithm</i>.

Let G be a finite graph with vertex set V and edge set E. A graph complex on G is an abstract simplicial complex consisting of subsets of E. In particular, we may interpret such a complex as a family of subgraphs of G. The subject of this thesis is the topology of graph complexes, the emphasis being placed on homology, homotopy type, connectivity degree, Cohen-Macaulayness, and Euler characteristic. We are particularly interested in the case that G is the complete graph on V. Monotone graph properties are complexes on such a graph satisfying the additional condition that they are invariant under permutations of V. Some well-studied monotone graph properties that we discuss in this thesis are complexes of matchings, forests, bipartite graphs, disconnected graphs, and not 2-connected graphs. We present new results about several other monotone graph properties, including complexes of not 3-connected graphs and graphs not coverable by p vertices. Imagining the vertices as the corners of a regular polygon, we obtain another important class consisting of those graph complexes that are invariant under the natural action of the dihedral group on this polygon. The most famous example is the associahedron, whose faces are graphs without crossings inside the polygon. Restricting to matchings, forests, or bipartite graphs, we obtain other interesting complexes of noncrossing graphs. We also examine a certain "dihedral" variant of connectivity. The third class to be examined is the class of digraph complexes. Some well-studied examples are complexes of acyclic digraphs and not strongly connected digraphs. We present new results about a few other digraph complexes, including complexes of graded digraphs and non-spanning digraphs. Many of our proofs are based on Robin Forman's discrete version of Morse theory. As a byproduct, this thesis provides a loosely defined toolbox for attacking problems in topological combinatorics via discrete Morse theory. In terms of simplicity and power, arguably the most efficient tool is Forman's divide and conquer approach via decision trees, which we successfully apply to a large number of graph and digraph complexes. / QC 20100622