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160 7 ⋅ conclusion Given any definition of PageRank, it must involve α, and it is surprising how little attention α received in early discussions. The first five years of PageRank research largely ignored the impact of α. That finally changed. It almost seemed as if α were “in the air,” to borrow a phrase from my late adviser. A stream of papers between 2003 and 2007 attacked α directly. These attacks established that PageRank is a rational function α [Boldi et al., 2005], examined the parametric structure of the so-called Google matrix definition of PageRank [Serra-Capizzano, 2005], and even proposed a theoretically motivated choice of α [Avrachenkov et al., 2007]—among other contributions, of course. This thesis fits the canon of that attack. We focus on the interaction be- tween α and PageRank from the perspective of sensitivity analysis in three ways. the derivative First, PageRank is a rational function of α. and a simple measure of the stability or sensitivity of such a function is the derivative. PageRank’s derivative with respect to α satisfies (I − αP)x′(α) = Px(α) − v. Chapter 3 begins with this derivative. We provide two theoretical con- tributions: a discussion of mathematical formulations of the PageRank derivative (section 3.1) and a theorem relating a first order Taylor step along the derivative to another PageRank vector (theorem 7). Further- more, we introduce a new algorithm to compute the derivative using any existing techniques to compute PageRank (section 3.2). random alpha Second, the random surfer model for PageRank on web pages suggests that the value of α ought to reflect the probability of real people following links when browsing the web. Chapter 4 embraces this view and follows it to its logical conclusion. Because PageRank is a nonlinear function of α, the PageRank vector with any aggregate probability α is incorrect and we need to regard α as a random variable distributed according to all the probabilities of following a link. In order to explore the resulting model computationally (section 4.6), we employ techniques from the uncertainty quantification community, which were developed to solve partial differential equations models with random variables. When α is random, the PageRank vector itself is also random, and the standard deviation of the random variables in the PageRank vector produces a new sensitivity measure for PageRankPDF Image | Instagram Cheat Sheet
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