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4.4 the random alpha pagerank model So far, we have discussed our vision for RAPr and the body of literature that surrounds our ideas. Now, RAPr is formally stated and analyzed. Given a random variable A with finite moments distributed within the interval [0, 1], the random alpha PageRank is the vector x(A) that satisfies (I − AP)x(A) = (1 − A)v (4.7) where I, P, and v are as in (2.5).11 When we use this model, we often look at E [x(A)] and Std [x(A)] . We address some theoretical implications of this model in the next few sec- tions, and we defer the discussion of computation until section 4.6 Remark 8. From this definition, we can immediately show that our model generalizes the TotalRank algorithm [Boldi, 2005], which produces a vector t defined as t= 1x(α)dα. 0 If A ∼ U[0, 1] in RAPr, then E[x(A)]= 1x(α)dα=t. 0 A purported benefit of the TotalRank algorithm is that it eliminated picking an α in a PageRank computation. When compared to RAPr, however, it corresponds to a particular choice of the random variable A. existence It behooves us to check that the expectation of RAPr is well defined. The concern is that E[x(A)] = ∫01 x(α)ρ(α)dα touches the value x(1) = 1. Looking only at the linear system (I − αP)x = (1 − α)v, we could conclude that x(1) does not exist because the matrix is singular when α = 1. If x(1) is not defined, then the expectation of RAPr will not exist. However, readers who thought this must have skipped a section in chapter 2. There is no difficulty for our formulation of PageRank because α = 1 corresponds with a removable singularity of the function.12 Thus, we can extend the definition of 11 Recall, I is the identity matrix, P is a column stochastic matrix (eT P = eT ), v is a non-negative probability vector (eT v = 1). x to α = 1 with the limiting value. In contrast, the quantity ∫ 1 y(α)ρ(α) dα 0 12 Recall section 2.7. We showed that x(1) uniquely exists and equals x(1) = XYv for X and Y based on the Jordan canonical form of P. 13 If the right-hand side vector in Pseudo- Rank is (1 − α)v, then y(1) is defined. does not exist for the PseudoRank vector y(α).13 This existence result is yet another reason that we prefer the PageRank definition to the PseudoRank definition. 4.4 ⋅ the random alpha pagerank model 63PDF Image | MODELS AND ALGORITHMS FOR PAGERANK SENSITIVITY
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