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26 2 ⋅ pagerank background 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 et al., 2008]. In terms of pure PageRank theory, a clear statement about the effect of P follows. Theorem4(Bianchinietal.[2005],theorem5.3). SupposePandPˆaretwo stochastic matrices of the same size. Given fixed α and v, the PageRank vectors for P and Pˆ are x and xˆ. Let U be the set of columns where P and Pˆ differ. Then 2α ∥ x − xˆ ∥ ≤ ∑ x i . 1−α i∈U Although the proof of this result is not long, it’s a diversion from the topic of this thesis.5 The gist is that what happens with P is relatively well studied. The same holds for v and even in the original PageRank paper, the impact of v is clear [Page et al., 1999]. Page et al. [1999] defined E = v and wrote: However, aside from solving the problem of rank sinks, E turns out to be a powerful parameter to adjust the page ranks. Intu- itively the E vector corresponds to the distribution of web pages that a random surfer periodically jumps to. As we see below, it can be used to give broad general views of the Web or views which are focussed [sic] and personalized to a particular indi- vidual. 4 The result in the article is slightly more general, but this statement is the motivation. 5 Although Gleich and Polito [2007] use this theorem with considerable bravura.PDF Image | ALGORITHMS FOR PAGERANK SENSITIVITY DISSERTATION
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