ALGORITHMS FOR PAGERANK SENSITIVITY DISSERTATION

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ALGORITHMS FOR PAGERANK SENSITIVITY DISSERTATION ( algorithms-for-pagerank-sensitivity-dissertation )

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Program 3 – Gauss-Seidel PageRank. This highly compressed implementation of Gauss-Seidel on a sub-stochastic matrix P = P ̄ T shows all steps necessary to compute (2.22) implicitly for the strongly or weakly preferential PageRank problem. It omits a few lines and is not “cut-and- paste” ready, but it retains the details in the essential pieces. See section 6.6 for information on where to get a complete implementation. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 (a) Gauss-Seidel iteration function [x flag reshist] = gspr(P,a,v,tol,maxit,verbose,u) % GSPR Compute PageRank with the Gauss Seidel algorithm % P: a row substochastic matrix; a=alpha; v=teleportation vector; % tol=stopping tolerance; maxit=max iterations; u=weakly personalized vector n = size(P,1); Ps=P; % make a copy so that gssweeppr computes extra info once x=zeros(n,1)+v; normed=true; extra=0; flag=0; delta=2; iter=0; reshist=zeros(maxit,1); t=0; z=0; dsum=[]; while itertol [x rdiff dsum Ps]=gssweeppr(x,Ps,v,a,(1-a),dsum,u); % rdiff is the difference if normed, nx = csum(x); x=x./nx; dsum = dsum./nx; end % evaluate the residual to make sure we are correctly converged if rdiff < tol, delta = prresid(x,P,a,v,dsum,u); extra = extra + 1; end if verbose, fprintf(’gs : m=%7i nm=%7i d=%8e c=%8e\n’, iter, ... iter+extra, delta, rdiff); end reshist(iter+1)=rdiff; iter=iter+1; end flag=delta>tol; reshist=reshist(1:iter); function delta=prresid(x,P,a,v,dsum,u) % compute the residual (I-aP)x - (1-a)v h = a*(P’*x); h = h + a*dsum*u + ((1-a)*csum(x))*v; delta = normdiff(h,x); (b) Gauss-Seidel sweep function [x ndiff dsumn Ps] = gssweeppr(x,P,v,a,g,dsum,u) if isstruct(P), n=P.n; ri=P.ri; cp=P.cp; id=P.id; Ps=P; else [cp ri]=sparse_to_csc(P); n=length(cp)-1; d=zeros(n,1); % compute CSC/inv degs for i=1:length(ri), d(ri(i))=d(ri(i))+1; end, d(d>0)=1./d(d>0); id=d; Ps = struct(’n’,n,’cp’,cp,’ri’,ri,’id’,id); % save for next iteration end if isscalar(u), uscalar=true; else uscalar=false; end if isscalar(v), vscalar=true; else vscalar=false; end dsumn1=0; dsumn2=0; ndiff1=0; ndiff2=0; vals=false; dsum1=dsum; dsum2=0; if isempty(dsum), dsum1=0; dsum2=0; % compute initial dangling sum for i=1:n, if id(i)==0, t=dsum1; z=x(i)+dsum2; dsum1=t+z; dsum2=(t-dsum1)+z; end,end end for end cpi=cp(i):cp(i+1)-1 j=ri(cpi); if vals, pji = ai(cpi); else pji=id(j); end if i==j, pii = pji; continue; end xn=xn+x(j)*pji; % Step 1 handle Pbar dsums = (dsumn1+dsumn2+dsum1+dsum2); if uscalar, xn=xn+dsums*u; ucurr=u; else xn=xn+dsums*u(i); ucurr=u(i); end if id(i)==0, xn = xn - x(i)*ucurr; pii = pii + ucurr; end % if vscalar, vi = v; else vi=v(i); end % xn=(a*xn+g*vi)/(1-a*pii); if id(i)==0 % page i is dangling Step 3 update x Step 4 update dsum t=dsum1; z=-x(i)+dsum2; dsum1=t+z; dsum2=(t-dsum1)+z; t=dsumn1; z=xn+dsumn2; dsumn1=t+z; dsumn2=(t-dsumn1)+z; end xi=x(i); if xn>xi, dxi = xn-xi; else dxi = xi-xn; end t=ndiff1; z=dxi+ndiff2; ndiff1=t+z; ndiff2=(t-ndiff1)+z; x(i)=xn; end dsumn = dsumn1+dsumn2; ndiff = ndiff1+ndiff2; % finalize sums % Step 2, u*d’ % Step 5 accumulate % the difference % between iterations % Step6setx 2.5 ⋅ pagerank parameters 27 for i=1:n xn=0; pii=0;

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