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50 3 ⋅ the pagerank derivative 3.3.3 Limiting derivatives Previously, section 2.7 established the PageRank vector when α = 1. Now, we differentiate the explicit PageRank function to establish the PageRank derivativewhenα=1.GivenP=XJX−1 withJ=[I X−1 = [ Y0 Y1 ]T , then we have I 16Inthelastchapter,J=[ D1 J2]for a diagonal D1 with all simple eigen- values of ∣λ∣ = 1 and J2 with all other eigenvalues. Thus, J1 = [ D1 J2 ]. 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 ],16 X=[X0 X1 ],and x(α) = X0Y0v + (1 − α)X1(I − αJ1)−1Y1v and x′(α) = (α − 1)X1(I − αJ1)−1J1(I − αJ1)−1Y1 − X1(I − αJ1)−1Y1v. This result matches Langville and Meyer [2006a, theorem 6.1.3], but with an explicit form for the group inverse of (I − P) using the Jordan form of P. 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. 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. J1 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 | ALGORITHMS FOR PAGERANK SENSITIVITY DISSERTATION
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