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Table 4.4 – PageRank vs. random alpha PageRank sensitivity on a big graph. The truncated τ values (τε (y, z) with ε = 10−10 ) again show that the standard deviation vectors produce different rankings from the expectation vectors for the graph uk-2006 with 77 million vertices and 2.2 billion edges. A1 is a Beta(2, 16, [0, 1]) random variable with statistics computed using a 25-point quadrature rule, and the parameter A2 is a Beta(1, 1, [0, 1]) random variable computed using a 10-point quadrature rule. The coloring is the same as in table 4.3. z 4.8 ⋅ applications 89 y x(0.5) x(0.85) x(0.95) E[x(A1)] E[x(A2)] Disjoint 1 0.8 0.6 0.4 0.2 x(0.85) x(0.95) E [x(A1 )] E [x(A2 )] Std [x(A1 )] Std [x(A2 )] 0.850 0.765 0.845 0.956 0.412 -0.538 0.910 0.967 0.891 0.294 -0.675 0.916 0.808 0.219 -0.706 0.892 0.287 -0.675 0.378 -0.577 Std[x(A )] vs. x(0.85) 1 Std[x(A )] vs. x(0.5) 2 Identical 0 10 10 10 10 k 0246 The second comparison metric is the intersection similarity metric [Boldi, 2005]. Given two ordered sequences of items A and B, let Ak (resp. Bk ) be the top k items in A (resp. B). Then 1 k ∣Aj∆Bj∣ isimk (A, B) = ∑ , (4.45) k j=1 2j where ∆ is the symmetric difference operator between two sets. The intersec- tion similarity is the average of the normalized symmetric differences for all top-j lists with j ≤ k. If the two orderings are identical, then isimk = 0 for all k. If the two sequences have disjoint items, then isimk = 1. Figure 4.13 displays this value for the standard deviations vectors. For A1, the intersec- tion similarity hovers around 0.3 with increases at 10, 1,000 and 10,000,000 pages. In contrast, Std[x(A2)] has a higher intersection similarity for the first 106 pages and orders the tail quite differently, resulting in a peak past 106 pages. This final peak is perhaps indicative of the negative τ correlation between Std [x(A2 )] and x(0.5). These results support our claim that the standard deviation of RAPr reveals characteristically new information for the underlying graph. Figure 4.13 – Intersection similarity be- tween PageRank and the RAPr model. The intersection similarity metric for the uk-2006 graph shows that the standard deviation vector is unlike the PageRank vector under this mea- sure. The computations were done for A1 ∼ Beta(2, 16, [0, 1]) with a 25-point quadrature rule and for A2 ∼Beta(1,1,[0,1])witha10-point quadrature rule. Intersection Similarity (k)PDF Image | MODELS AND ALGORITHMS FOR PAGERANK SENSITIVITY
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