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To solve the PageRank linear system, do not compute A = (I − αP) and apply the iteration. Instead, proceed implicitly, just like for the power method. With just P ̄, working implicitly is easy; with P, it is hard. The law codes for PageRank [Vigna et al., 2008] describe the implicit implementation in some detail. In is implemented in the not-quite-complete program 3. Arguably the key step is separately accumulating dT x on the current iterate (dsum) and the updated entries (dsumn). Beyond that detail, the implementation is straightforwardly an implicit version of (2.22). But let’s outline the update at each iteration anyway! For Gauss-Seidel, there is no large benefit to working with strongly vs. weakly preferential PageRank and so the following iteration actually computes the update for the weakly preferential PageRank formulation. That is, A = (I − αP ̄ − αudT ). Suppose that θ = dT x before any changes to x, and that θ ̄ = dT x after we change an element of x. Let x ̄i be a temporary update value and di = 1 if page i is dangling. To update xi , the steps follow. 1. Computeτ=∑j=/i P ̄ijxj. 2. Add (θ + θ ̄)ui to τ but subtract ui if page i is a dangling page. 3. Finalize the value of x ̄ = ατ+(1−α)vi . i 1 − α P ̄ i i − α u i d i 4. If page i is dangling, subtract xi from θ and add x ̄i to θ ̄. 5. Accumulate the difference between xi and x ̄i . 6. Setxi =x ̄i. These steps correspond to the labeled steps in the Gauss-Seidel sweep func- tion. Gauss-Seidel has a few subtleties. Suppose P has no diagonal entries, then the power method, Richardson, and Jacobi iterations all coincide. Although the asymptotic performance of Gauss-Seidel is better than that of the Jacobi method, a simple test shows that this need not hold for the average perfor- mance. n = 10000; rand(’state’,0); % setup a 10000x10000 graph with adjacency matrix A A = sprand(n,n,10/n); A = spones(A); A = A - diag(diag(A)); P = normout(A); % normalize the out-degrees alpha = 0.99; % exacerbate the issue. [xp,flag,histp] = powerpr(P,alpha); [xp,flag,histgs] = gspr(P,alpha); fprintf(’power method takes %i iterations\n’, length(hist)); fprintf(’gauss seidel takes %i iterations\n’, length(histgs)); 2.4 ⋅ algorithms 29PDF Image | Instagram Cheat Sheet
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