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82 4 ⋅ random alpha pagerank 4.6.3 Path damping As discussed in sections 4.3.3 and 4.4.3, path damping algorithms for Page- Rank are not novel. RAPr simply provides a large set of functions that gener- ate the path damping coefficients. In this section, we will discuss using these ideas to compute E [x(A)] and Std [x(A)]. Recall the Neumann series from theorem 9, ∞ E[x(A)]=∑E[Al −Al+1]Plv. (4.21) l=0 If we truncate this series to a finite value N, then an algorithm for E[x(A)] immediately follows: NN E[x(A)]≈x(N) =∑E[Al −Al+1]Plv+(1−∑E[Al −Al+1])PN+1. l=0 l=0 (4.22) The final term in this summation ensures that eT x(N) = 1 for the path damp- ing approximation. To compute Std [x(A)] using the path damping equations we compute E [x(A) ● x(A)] and then compute √ Std [x(A)] = Based on the Neumann expansion, E [x(A) ● x(A)] − (E [x(A)] ● E [x(A)]). ∞∞ E[x(A)●x(A)]=∑∑E[Ai+j −2Ai+j+1 +Ai+j+2](Piv)●(Pjv). (4.23) i=0 j=0 And again, we truncate this series to a common term in both i and j: N E[x(A)2]≈s(N) =∑E[Ai+j −2Ai+j+1 +Ai+j+2](Piv)●(Pjv). (4.24) i,j Note that we do not apply any correction to the sum to ensure a summation property of the solution as in the case for E [x(A)]. Given the moments of the distribution A, μk (A) = E [Ak ] , 0 ≤ k ≤ 2N + 2, (4.25)PDF Image | Instagram Cheat Sheet
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