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8 1 ⋅ introduction A simple approach to investigating the sensitivity of the PageRank vector is to look at the derivative with respect to α, which we explore in chapter 3.8 Algorithms to compute the derivative of PageRank were already known, but we propose a new algorithm that can use any existing PageRank solver without modification. The results from our algorithm (algorithm 1) on the pages from Wikipedia are shown in table 1.2. 8 Although PageRank was described as a random surfer model, it also has a nice expression as a function of the parameter α. Our investigation is of the derivative of this function with respect to α. α = 0.50 United States C:Living people United Kingdom Race in the US. Census C:Ctgs. by country France England Canada Germany World War II List of sovereign states α = 0.85 C:Main topic classif. C:Contents C:Fundamental C:Wikipedia admin. C:Ctgs. by topic C:Society Por:List of portals C:Articles Por:Contents/Portals C:Ctgs. by location C:Categories α = 0.99 C:Main topic classif. C:Contents C:Fundamental C:Wikipedia admin. C:Ctgs. by topic C:Society Por:List of portals C:Articles Por:Contents/Portals C:Ctgs. by location C:Ctgs. by country After investigating the derivative, we develop a new model for PageRank in chapter 4 along with a significant new approach to sensitivity analysis. Instead of using a derivative, which just measures the effect of small change in α, this new model examines an approach based on the variance of the PageRank vector as a function of its parameter over a wide range of values of α. Another interpretation shows that this sensitivity measure corresponds to replacing α in PageRank with a random variable and studying the standard deviation of a set of random PageRank variables. Hence, we call our new method RAPr— Random Alpha Pagerank. Figure 1.3 illustrates how sensitivity works in our new model. Our best-performing algorithm on this problem only involves computing PageRank vectors. x x 2 x 3 x x 5 x 6 0 0.5 Table 1.2 – Pages in Wikipedia with the largestderivative. PagesinWikipedia with the largest derivative, by value, not by magnitude. 1 Figure 1.3 – PageRank with a random vari- able as the teleportation parameter. This plot shows how likely each PageRank value between 0 and 0.5 is when the constant parameter α is replaced with a random variable. Rather than computing the value of PageRank at the circle stem points, we look at the entire range of values it might take. The width of the major portion of the curves is the new sensitivity parameter. See chapter 4 for more information about this plot and the details of the model. 4PDF Image | ALGORITHMS FOR PAGERANK SENSITIVITY DISSERTATION
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