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30 2 ⋅ pagerank background PageRank and therefore not cheap. Also Boldi et al. [2005] explored using the derivatives to extrapolate the PageRank vector to values of α close to 1. Meanwhile, people continued to work on what happens when α → 1. Their efforts are useful: we use their results in this thesis. Although it is wrong for the web it is mathematically interesting and the story as α → 1 is our next topic. counterpoint The preceding discussion makes simplifying assump- tions—it is a modeling argument. From it, we conclude that taking α close to 1 is not a good idea if the goal is to produce a useful ordering of web pages. There are other goals, and we do not mean to imply that PageRank computations with α near 1 are entirely useless. Indeed, in chapter 4, we use computations with α close to 1 inside a variation on the PageRank model. Furthermore, the argument gave no practical guidance about when α is too close to 1 beyond the simple advice α = 0.5. Our point is simply that setting α large should be considered carefully. Using a small α (0.5 − 0.9) is not a mere matter of computational convenience, there are important reasons why it should be so. 2.7 the limit case For all α < 1, the PageRank vector is unique. Yet there may be many x that satisfy Px = x (the PageRank equation when α = 1). From the previous section, PageRank is a rational vector function of α, so what happens when α = 1? The limit exists! That is, lim x(α) α→1 exists and is unique.7 2.7.1 The linear system Looking at (I−αP)x = (1−α)v is the easiest way to find the limit. Consider the Jordan canonical form P = XJX−1 . Because P is a stochastic matrix, all the eigenvalues λ that have ∣λ∣ = 1 are semisimple [Meyer, 2000, page 696] 7 For trivial loop-only graph 12345 P=I and the PageRank vector x(α) = v for all α < 1. The limit vector is also v but any vector satisfies Px = x (the PageRank equation when α = 1). and thus I ⎡⎤ ⎢⎥ J = ⎢ D1 ⎥ , ⎢ J ⎥ ⎣ 2⎦ (2.24)PDF Image | ALGORITHMS FOR PAGERANK SENSITIVITY DISSERTATION
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