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58 3 ⋅ the pagerank derivative 3.4 experiments Finally, we study the predictive power of the PageRank derivative. 3.4.1 Does a negative derivative justify a change in ranking? One of the most promising uses of the derivative vector is to evaluate what happens in the PageRank vector at different values of α. Table 3.2 shows some results on this idea where we look at the fraction of pages with negative deriva- tive that actually decrease in rank when α increases by a value γ. The fraction predicted by the derivative is higher than the average fraction predicted by a random vector. We do not consider the magnitude of the derivative with these predictions. These results are mixed. For large values of γ, cnr-2000 shows a marked increase in predictive power using the derivative over a random vector. On the Wikipedia graphs, in contrast, there is almost no difference between using a random vector and the derivative. 3.4.2 What are the pages with the largest derivative? For the largest strongly connected component of wiki-2006-11, table 3.3 lists the top 20 pages with largest derivative for a few values of α. Most of the pages that appear in the top 20 list are also highly ranked according to the PageRank value. Additionally, pages in the “category” namespace in Wikipedia are highly ranked by both PageRank and its derivative for the two largest values of α evaluated. Table 3.2 – Prediction of rank change with the derivative. The x′ entries show the fraction of pages with negative derivative that decreased in rank when α is increased by the value of γ in the table heading. These values are compared with the r entries, which show the average fraction over 50 trials in which the derivative is replaced by a random vector generated with randn in Matlab. For aa-stan, the ranking did not change and thus all predictions were incorrect. Graph aa-stan ee-stan cs-stan cnr-2000 wiki-2006-09 wiki-2006-11 = 0.001 = 0.01 x′ r x′ r = 0.1 x′ r 0.000 0.000 0.478 0.453 0.505 0.432 0.641 0.553 0.361 0.342 0.360 0.341 0.000 0.000 0.079 0.078 0.257 0.237 0.557 0.477 0.385 0.362 0.385 0.361 0.000 0.000 0.286 0.266 0.441 0.372 0.621 0.527 0.385 0.362 0.383 0.360PDF Image | Instagram Cheat Sheet
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