Linear Algebra and its Applications

Sep Kamvar’s monograph presents work from his 2004 thesis on how numerical algorithms are a critical component in search systems. Kamvar focuses on the new information networks arising in the late 1990s and early 2000s. When originally published, these results were groundbreaking and resulted in an NSF press release and even a Slashdot listing. 2 The book is divided into two parts, one for each of the two types of distributed networks discussed: the world-wide web (Part 1) and peer-to-peer file sharing networks (Part 2). Chapter 1 includes a brief synopsis of these two parts; indeed, it is much like the following two paragraphs of this review. Part 1 focuses on the PageRank algorithm, a technique used by the Google search engine to evaluate the quality of a web-page. Chapter 2 tersely introduces the overall PageRank idea, the underlying PageRank Markov chain, and how the power method computes the PageRank vector, which is the stationary distribution of the PageRank Markov chain. Chapter 3 presents a theorem on the eigenvalues of the transition matrix of the PageRank Markov chain – the so-called Google matrix. Using these eigenvalues, the next chapter analyses the condition number of a linear system to compute thePageRankvector. Chapters5–7develop threedifferent techniques toaccelerate computing thePageRankvectors. Thefirst technique (Chapter5) is anextensionof classic extrapolationapproaches such as Aitkin extrapolation and quadratic extrapolation, albeit customized to the PageRank problem, which reduces the wall-clock time by 30%. The second technique (Chapter 6) capitalizes on the different convergence behaviors of individual components of the PageRank vector. Some components converge after only a few iterations and eliminating computation associated with these components reduces the overall computation time. The final technique (Chapter 7) exploits block structure in the PageRank matrix of the web-graph to develop a multi-level approach. Kamvar’s goal with these three techniques is to accelerate PageRank to enable a scalablepersonalizedPageRankapproach,where there is a PageRank vector for each user of the search engine. At the end of this chapter, he discusses how BlockRank achieves this goal. Part 2 tackles search in peer-to-peer file sharing networks, like those used in the originalbut-now-defunct Napster and Kazaa applications. In such systems, queries for information are propagated among connected clients. After a query produces a result, the clients form a new connection and transmit the information. Everything is decentralized in these networks. Consequently, reputable behavior, which is accurately responding to queries and file transmission requests, is a prized com-