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of the variability in the PageRank induced by the variability in A. For the graph in figure 4.2, the standard deviation vector is Std [x(A)] = ⎢ ⎢ This vector shows that x5 and x6 have the highest standard deviation. In a traditional PageRank context, these pages are both in a sink-component and accumulate rank from the largest connected component (x1 , x2 , and x4 ). A high standard deviation signals that the rank of these pages is “more likely” to change for different realizations of A. Another interesting quantity derived from our model is the correlation coefficient between ranks. The correlation coefficient between two ranks xi and xj provides a measure of how xi will vary as xj varies with different realizations of A. If the correlation between xi and xj is positive then an increase in x i from separate realizations of A implies that x j tends to increase as well. If the correlation is negative, then an increase in xi implies a decrease in x j is likely. Here, we have computed the correlation coefficient between all pages:3 ⎡⎢ 0.021 332 ⎤⎥ ⎢⎥ ⎢ 0.019 883 ⎥ ⎢⎥ ⎢ 0.026 146 ⎥ ⎥ . 0.023 193 ⎥ ⎢⎥ ⎢ 0.041 233 ⎥ ⎢⎥ ⎢⎣ 0.049 304 ⎥⎦ ⎡⎢ 1.000 000 0.999 996 0.998 844 0.999 211 −0.999 951 −0.999 373 ⎤⎥ ⎢⎥ ⎢ 0.999 996 1.000 000 0.998 764 0.999 149 −0.999 936 −0.999 313 ⎥ ⎢⎥ ⎢ 0.998 844 0.998 764 1.000 000 0.999 963 −0.999 261 −0.999 920 ⎥ ⎢⎥. ⎢ 0.999 211 0.999 149 0.999 963 1.000 000 −0.999 550 −0.999 989 ⎥ ⎢⎥ ⎢ −0.999 951 −0.999 936 −0.999 261 −0.999 550 1.000 000 0.999 667 ⎥ ⎢⎥ ⎢⎣ −0.999 373 −0.999 313 −0.999 920 −0.999 989 0.999 667 1.000 000 ⎥⎦ The sign pattern in the correlation structure shows that there are effectively two groups of pages, (x1, x2, x3, x4) and (x5, x6). Thus far, we have seen a few useful quantities that we can derive from the RAPr model. In practice, we anticipate many problem-dependent uses for these quantities. Our experiments show that the standard deviation vector is uncorrelated (in a Kendall-τ sense4) with the PageRank vector itself. Because pages with a high standard deviation have highly variable PageRank values, 4 Kendall’s τ difference in concor- dant and discordant pairs between two lists relative to an identical ordering and an inverted ordering. The τ value is 1 for identical lists and −1 for inverted lists. 4.2 ⋅ vision 67 3 This matrix is symmetric, so we could simply present one half of it.PDF Image | Instagram Cheat Sheet
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