Browse Books

Reviewer: Esfandiar Haghverdi

Graphs are among the most successful abstract models in engineering, computer science, the social sciences, and the natural sciences. However, as the problems and challenges in such domains grow in complexity, graph algorithms need to keep pace. To improve the computational efficiency of graph algorithms, researchers have proposed a shift to a parallel computing paradigm. This monograph is an edited volume collecting contributions by more than 20 researchers in the field where challenges of parallel graph algorithm implementation and good parallel performance issues are addressed by exploiting the duality between the canonical representation of graphs as a collection of vertices and edges and a sparse adjacency matrix representation. Such a linear algebraic approach allows for a wider pool of applications and complements the extant body of literature on graph algorithms. Moreover, a linear algebraic approach offers syntactically more compact algorithms that are easier to implement and can be more easily optimized to yield higher performance. The book is divided into three parts: "Algorithms," Part 1, presents the basic mathematical framework; "Data," Part 2, provides a number of examples where a linear algebraic approach is used to develop new algorithms for modeling and analyzing graphs; and "Computation," Part 3, focuses on sparse matrix computation. Over 40 algorithms are presented; even though most of these are presented in pseudocode (when working code examples are required), these are expressed in MATLAB. Beginning Part 1, chapter 1 briefly surveys some of the more interesting results found later in the book. In particular, graph/adjacency matrix duality is discussed. Chapter 2 presents the notation, definitions and conventions for graphs, matrices, arrays, and operations on them to be used throughout the remainder of the book. Chapter 3 presents two different examples of linear algebraic graph algorithms: strongly connected components are obtained via efficient computation of infinite powers of the adjacency matrix, and shortest paths are computed using a modification of matrix exponentiation. Chapter 4 provides some of the foundations of linear algebraic graph algorithms together with a number of classic algorithms, using MATLAB syntax. Chapter 5 discusses the representation of single-source shortest paths, all-pairs shortest paths, and minimum spanning tree algorithms as algebraic operations. Chapter 6 continues with algebraic representation for more complex graph algorithms: clustering, vertex betweenness centrality, and edge betweenness centrality. Chapter 7 extends the algebraic model from matrices to tensors in order to model multi-link graphs. Various experiments are presented, including distinguishing between papers written by different authors with the same name and predicting the journal in which a paper is published. Chapter 8 combines hidden Markov models and Kronecker graphs to provide estimates of the signal-to-noise ratio, probability of detection, and probability of false alarm for different classes of vertices in the foreground for the subgraph detection problem. This approach is then applied to the problem of detecting a partially labeled tree in a power law background graph. Part 2 opens with chapter 9, which includes a description of a generative model for networks that is mathematically tractable and produces the common properties of real-life networks. The model uses the Kronecker product to generate graphs called Kronecker graphs. The properties of such graphs are studied in great detail and accompanied by many experimental results. Chapter 10 follows the discussion in the preceding chapter and offers an analytical theory of power law graphs based on the Kronecker graph generation technique, including explicit, stochastic, and instance Kronecker graphs. Chapter 11 takes on the challenge of visualization of Kronecker graphs, describing an interactive framework to assist scientists and engineers in generating, analyzing, and visualizing Kronecker graphs with as little effort as possible. Part 3 starts with chapter 12, which discusses several new results related to large-scale graph analysis using centrality indices. The first parallel algorithms for evaluating such computationally intensive metrics are given. The chapter also includes an application of betweenness centrality analysis to eukaryotic protein interaction networks. Chapter 13 reviews and evaluates a number of storage formats for sparse matrices and their impact on primitive operations. Chapter 14 presents the first parallel algorithms that achieve increasing speedups for an unbounded number of processors. Chapter 15 discusses automated techniques for mapping algorithms onto parallel architectures, and chapter 16 concludes the book with a discussion of some of the questions pertaining to the mathematical and computational foundations of the analysis of large graphs. Despite being a collection of contributions by various authors, the editors and authors have taken great care to present a coherent work on the topic, thus making it suitable as the main textbook for a course on linear algebraic graph algorithms. This book can also be used as a main or auxiliary textbook for a course on graph algorithms for engineers and scientists outside the field of computer science. Online Computing Reviews Service