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74 4 ⋅ random alpha pagerank Given the moments of the distribution A, μk (A) = E [Ak ] , 0 ≤ k ≤ 2N + 2, (4.25) the previous summation expressions become algorithms. As discussed in sec- tion 4.4.1 we only consider A ∼ Beta(a, b, [l , r]). For A ∼ Beta(a, b, [0, 1]), the values μk are known analytically [Zwillinger et al., 1996]: b+k k b+j μ0 = 1, μk = To handle the general case, define μˆj ≡μj(A)whereA∼Beta(a,b,[0,1]). For A ∼ Beta(a, b, [l , r]), ((r−l)τ+l)kρ(0,1) Beta(a,b) E[Ak]= = ∑ ( ) μˆ j ( r − l ) l Beta(a,b) l0 kk jk−j j=0 j (4.28) and we can compute the moments of A ∼ Beta(a, b, [l , r]) by scaling and shifting those of A ∼ Beta(a, b, [0, 1]). Program 6 gives a simple implementa- tion of the path damping algorithms and an implementation of the recursion (4.29) μk(A) = μ(0,k) j j−i μ(i,j) = ∑( )μˆ (r−l)m−ilj−m =(r−l)μ(i,j−1) +lμ(i+1,j) m=i m−i m 18 The implementation is not straightfor- ward, though it is correct. It follows from organizing the moments into a matrix ⎡⎢ μ(0,0) μ(0,1) ... μ(0,k) ⎤⎥ ⎢ μ(1,1) ... μ(1,k) ⎥ ⎢ ⋱⋮⎥ ⎣ μ(k,k) ⎦ and filling in the entries μ(0,1), . . . , μ(0,k) from the initially specified diagonal. At every step in the implementation, we compute a new diagonal. to compute the moments μk (A).18 k ≥ 1. a+b+k+1 μk−1 = ∏ , j=1 a+b+j+1 (4.26) (4.27) (τ)dτ ζkρ(l,r) (ζ)dζ = r 1PDF Image | MODELS AND ALGORITHMS FOR PAGERANK SENSITIVITY
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