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32 2 ⋅ pagerank background 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. Manipulating v begets the personalized or topic specific PageRank variants alluded to by chapter 1. These problems have been extensively studied [Haveli- wala, 2002; Jeh and Widom, 2003] and personalized PageRank forms the basis of novel clustering algorithms [Andersen et al., 2006] and (loosely in- terpreted) interpolation schemes for graphs [Zhou et al., 2005]. That leaves α, which merits its own section. 2.6 the pagerank function of the damping parameter Much of the initial research on α is motivated by the idea expressed in the following statement [Langville and Meyer, 2006a, page 58]: But the larger values of α are the ones of most interest because they give more weight to the true link structure of the Web while smaller values of α increase the influence of the artificial proba- bility vector vT . Since the PageRank concept is predicated on tak- ing advantage of the Web’s link structure, it is natural to choose α closer to 1. 5 Although Gleich and Polito [2007] use this theorem with considerable bravura.PDF Image | Instagram Cheat Sheet
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