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86 4 ⋅ random alpha pagerank 4.8 applications Thus far, we have theoretically examined the RAPr model, given algo- rithms to compute its statistics, and analyzed those algorithms; we have yet to address applications of this model. While the expected value of the RAPr model appears to order nodes like the deterministic PageRank vector at the expected α, the standard deviation vector orders nodes differently. We first demonstrate this behavior for a range of graphs and distributions of A. Then, we show similar observations on a large web graph and discuss the inter- section similarity of the standard deviation vector for this graph. Next, we present an example of our model outside the web graph domain and ob- serve that this ranking behavior of the standard deviation vector holds for a gene ranking application. Finally, we show that using the standard deviation information aids a spam classification task. 4.8.1 PageRank To begin our empirical analysis of RAPr, we present table 4.3. For the four Beta distributions we have examined throughout this chapter, the table presents the 1-norm, Kendall’s τ correlation coefficient, and a truncated-τ correlation coefficient between x(E [A]), E [x(A)], and Std [x(A)]. The 1- norm difference is rescaled to be related to a correlation coefficient when applied to probability distribution vectors. The truncated-τ or τε measure removes digits less than ε before computing τ. Formally, τε (y, z) = τ(ε round(y/ε), ε round(z/ε)), (4.44) where the “round function” rounds to the nearest integer. The τε measure is motivated by inconsistencies with the τ measure and inaccurate computa- tion [Boldi et al., 2007]. The expectation and standard deviation were com- puted with a 33-point quadrature rule28 and each PageRank system solved to a weighted 10−9 tolerance (see section 4.6.4). From the table, we observe: • thePageRankvectorx(E[A])andtheexpectedvalueintherandom model E [x(A)] are numerically similar and induce similar orderings of the pages; • thestandarddeviationvectorStd[x(A)]isneithernumericallysimilar nor similar in either τ metric to x(E [A]); • using τε can give different results; and • thebehaviorofthestandarddeviationvectorisnotconsistentbetween graphs and distributions. The first shaded column group of the table justifies the first statement. The marked reduction in shading in the second column group explains the sec- ond, and the seemingly random values in this column group justify the last statement. Interestingly, four graphs behave nearly the same: uk-2006-host, uk-2007-host, eu-2005, and us2004cc. With the exception of uk-2007-host, 28 See note 27 for a comment about the number of points.PDF Image | MODELS AND ALGORITHMS FOR PAGERANK SENSITIVITY
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