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Disjoint 1 0.8 0.6 0.4 0.2 Figure 4.13 – Intersection similarity between PageRank and the RAPr model. The intersection similar- ity metric for the uk-2006 graph shows that the standard devia- tion vector is unlike the Page- Rank vector under this measure. 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]) with a 10-point quadrature rule. Identical 0 10 10 10 10 k 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. 4.8.3 Gene regulatory networks Recently, many authors have used PageRank-type equations as measures on arbitrary graphs. Among these measures are GeneRank [Morrison et al., 2005] for identifying important genes in a regulatory network, ProteinRank [Freschi, 2007] for identifying important proteins, and IsoRank [Singh et al., 2007] for identifying important edges in a graph-isomorphism-like problem. We will demonstrate the results of RAPr on the GeneRank problem using the data published for that paper. 0246 4.8 ⋅ applications 99 Std[x(A )] vs. x(0.85) 1 Std[x(A )] vs. x(0.5) 2 Intersection Similarity (k)PDF Image | Instagram Cheat Sheet
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