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88 4 ⋅ random alpha pagerank Theorem13. IfA∼Beta(a,b,[0,1])withintegersa>0andb>0,then approximating E [x(A)] with an N -sample Monte Carlo method takes logε a a N ∑(−1)k( )log(b+i+1) (4.38) Beta(a+1,b+1) k=0 k matrix multiplications. One problem with this theorem is that it does not handle a = 0, b = 0—the case when A is uniformly distributed. Computing this expectation exactly 1 −1 is impossible in this case because the indefinite integral ∫0 log (τ) d τ does not converge. To handle this case, we would need to quantify the convergence of the power method when α is bigger than the second largest magnitude eigenvalue of P. More precisely, P has many eigenvalues on the unit circle. Let λ2 be the magnitude of the first eigenvalue inside the unit circle. We need the convergence of the power method when α > λ2 and that will depend more strongly on λ2 than on α. 4.7.2 Path damping When A ∼ Beta(a, b, [0, r]), r ≤ 1, we can explicitly bound the conver- gence of the path damping algorithm for E [x(A)]. Recall the path damping approximation from (4.22): NN E[x(A)]≈x(N) =∑E[Al −Al+1]Plv+(1−∑E[Al −Al+1])PN+1v. Note that Thus we have N (1−∑E[Al −Al+1])=E[AN+1]. l=0 (4.39) l=0 l=0 ∞ ∥x(N)−x⋆∥=∥E[AN+2]PN+1v− ∑ E[Al−Al+1]Plv∥ l=N+2 ∞ ≤E[AN+2]+eT ∑ E[Al −Al+1]Plv l=N+2 ≤ 2E[AN+2]. We could have removed the final normalization term E[AN+1]PN+1v in the summation and bounded the result by E[AN+1] instead. However, the iter- ation we outlined in the algorithms section gives us better performance in practice.22 22 Give it a try yourself; maybe you will have a different experience.PDF Image | Instagram Cheat Sheet
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