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and Vigna, 2004] through a custom version of bvgraph or libbvg software (section 6.5). 4.6.2 Monte Carlo An enticingly straightforward method to compute the expectations, stan- dard deviations, and density functions of RAPr is to use a Monte Carlo method. To wit, first generate N realizations of A from a chosen distribu- tion, and then solve each resulting PageRank problem. With the N different realizations of x(αi), i = 1, . . . , N, we can compute unbiased estimates for E [x(A)] and Std [x(A)] with the formulas E[x(A)] ≈ S t d [ x ( A ) ] ≈ 1N ∑x(αi) ≡ μˆx, N i=1 N ∑ ( x ( A ) − μ ˆ ) 2 Unfortunately, as with any Monte Carlo method, these estimates con- √ verge as 1/ N [Asmussen and Glynn, 2007], which makes this approach prohibitively expensive for large graphs such as the web graph. The real advantage of the Monte Carlo method is its beautiful simplicity. The following short code is our entire implementation of the Monte Carlo method, including a numerically stable method to update the running vari- ance computation [Chan et al., 1983]. Program 5 – Computing RAPr with Monte Carlo. A Monte Carlo code in Matlab to estimate the expectation and standard deviation of the RAPr model. from Asmussen and Glynn [2007]. N − 1 i=1 1 ix 4.6 ⋅ algorithms 81 1 2 3 4 5 6 7 8 9 10 11 function [ex,stdx] = mcrapr(P,N,ba,bb,bl,br) tol=1e-9; maxterms=500; n=size(P,1); v=1/n; alphas = betarnd(bb+1,ba+1,N,1)*(br-bl) + bl; ex=zeros(n,1); stdx=zeros(n,1); for i=1:N % solve the PageRank system x = inoutpr(P,alphas(i),v,tol,2*ceil(log(tol)/log(alphas(i)))); % update the running solution sum and variance sum formulas ex = ex+x; if i>1, stdx = stdx + (1./(i*(i-1))).*(i*x-ex).^2; end end ex = ex./N; stdx=sqrt(stdx./(N-1)); % compute the mean and stdPDF Image | Instagram Cheat Sheet
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