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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, these graphs have the highest percentage of nodes in the largest strong com- ponent. The graph uk2005 demonstrates the largest discrepancy between τ and τε. This relatively large difference may signify that it differs characteristically from the other graphs. However, most of its standard deviation values are less than 10−8, so truncating the τ metric with ε = 10−8 may lose important information. Another explanation for the discrepancy is that more than half of the nodes in this graph have no links. 4.8.2 PageRank on a large graph The graphs in the previous section are small compared with the size of the true web graph. Now we address computing the quantities on a graph with 78 million nodes and just under 3 billion edges: the uk-2006 web spam test graph [Castillo et al., 2006].29 Our distributions of interest are A1 ∼ 29 Even this graph is tiny compared Beta(2, 16, [0, 1]) and A2 ∼ Beta(1, 1, [0, 1]). We chose the former because E[A1] = 0.85, the canonical value of α, and the latter because E[A2] = 0.5, a recently proposed alternative value of α. Both of these distributions have small a and support that extends all the way to 1. This makes computing the solution with path damping a difficult proposition, so we choose to use Gaussian quadrature. For A1 we used a 25-point rule, and for A2 we used a 10- point rule. The error bounds on quadrature state that these results may have with the real web graph. 4.8 ⋅ applications 97PDF Image | Instagram Cheat Sheet
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