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small set of αs. This idea is related to the Gauss quadrature algorithm of section 4.6.4. In RAPr, the random variable A has an associated quadrature rule for its expectation that specifies the αs at which to compute the function. RAPr is also more general. It is not tied to just computing a spam correlation but produces a correlation between any group of pages as we illustrated in section 4.2. 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, 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). 4.4 ⋅ the random alpha pagerank model 71PDF Image | Instagram Cheat Sheet
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