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28 2 ⋅ pagerank background 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. Simply put, the idea is that α < 1 is introducing a distortion into the rankings. As we will see, this sentiment is incorrect for the web.6 To look at what happens with α in PageRank, we study the implicitly defined PageRank function of α, 6 See the last paragraph of this section for another viewpoint. Also, we do not mean to suggest that such ideas were misguided. Newer research just provides better guidance. (I − αP)x(α) = (1 − α)v, which people sometimes write as x(α) = (1 − α)(I − αP)−1v. (2.23) Of course, we do not mean to suggest actually computing such functions explicitly and analyzing their properties, although, we are going to do so for purely expository purposes. For instance, 3 2 ⎢ 3 α3+6 α2−36 ⎥ 2 (α−1) (α+3) ⎥ ⎡1−α(α+3) ⎤ ⎢ 6 G = (α−1) (α+2) (α+3) ⎥ 3 α3+6 α2−36 ⎥ with v = e/6, yields x(α) = ⎢ 45⎢−⎥ 3 α3+6 α2−36 ⎥ ⎢⎥ ⎢ ⎢ ⎢ 1 (α+2)(α−4) ⎥. 16 ⎢6−(α+1)(α3+2α2−12)⎥ ⎢ 2 2 α3+4 α2−24 ⎥ ⎢⎥ ⎢ α3 7α2 5α ⎥ ⎢1 3+6+3 ⎥ ⎢⎥ ⎢α32 ⎥ ⎢− 6+α+4α+4 −1⎥ ⎣ (α+1)(α3+2α2−12) 6⎦ While this expression looks challenging to interpret, figure 2.5 shows the PageRank function of a single node, x1(α) on the 335-node largest strong component of the harvard500 graph. The expression in that figure looks nearly impossible to understand, and thus we need a more rational (pun fully intended) approach to the problem. Both of these examples were computed by the Matlab symbolic toolbox.PDF Image | ALGORITHMS FOR PAGERANK SENSITIVITY DISSERTATION
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