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30 2 ⋅ pagerank background The results are startling: Such results, however, are not common. Figure 2.3 shows typical behavior on web graphs. the problem Gauss-Seidel is a fast algorithm for PageRank. Its fatal flaw is that it requires access to P ̄ by rows, but standard data structures provide access to P ̄ by columns. While transposing a matrix in Matlab is as easy as Pt = P’, for a gigantic matrix it is not as easy. So Gauss-Seidel imposes some restrictions on the data structures. Another problem with Gauss-Seidel is that it cannot be parallelized effectively. Parallel variants of Gauss-Seidel exist [Saad, 2003] but they require a good multicoloring of the graph structure underlying the matrix to be effective. Thus it is not an appropriate algorithm for really large-scale problems. Nonetheless, it is often the best-performing serial algorithm. Use it when possible. 2.4.4 Summary Gauss-Seidel concludes our discussion of classic algorithms for PageRank. Algorithms for PageRank do not live with the problem (I − αP)x = (1 − α)v but at the higher level of a weakly or strongly preferential framework. Some algorithms even operate at the graph level. The PageRank problem requires these optimizations. See the discussion in section 5.1 for more about new algorithms developed for PageRank. As explained there, unfortunately, these new algorithms have little to recommend them over the classic power method and Gauss-Seidel iterations. 2.5 pagerank parameters Recall that the data for PageRank (problem 1) are P, v, and α. Varying these parameters often has a large effect on the PageRank vector x. Many of these effects are well understood. For example, when P comes from a graph in the strongly personalized PageRank model, then adding a new edge from node i to node j increases xi [Chien et al., 2004].4 Other results often focus on applications of link manipulation to increase PageRank values [Zhang et al., 2004; de Kerchove Figure 2.3 – Gauss-Seidel vs. the power method. On the ubc-cs graph with α = 0.85, the Gauss-Seidel method handily beats the power method. 0 10 −3 10 −7 10 0 20 40 60 80 100 Iteration 0 Power 10 Gauss−Seidel power method takes 25 iterations gauss seidel takes 304 iterations 10 −2 4 The result in the article is slightly more general, but this statement is the motivation. 0 10 20 ResidualPDF Image | Instagram Cheat Sheet
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