pagerank

% the set of pages that have a non-zero personalized PageRank but are not
% in the set of active pages. As options.bp -> 0, both of these
% approximations compute the actual personalized PageRank vector.
%
% pagerank.m
% David Gleich
%
%
% 21 February 2006
% -- added approximate PageRank
%
% Revision 1.10
% 28 January 2006
% -- added different computational modes and timing information
% options.x0: the initial vector [vector | {options.v}]
% options.alg: force the algorithm type
% ['gs' | 'power' | 'linsys' | 'dense' | {'arnoldi'} | ...
% p{4} is the vector computed from the 'linsys' algorithm with bicgstab
% p{5} is the vector comptued from the 'linsys' algorithm with gmres
% the other outputs all match these indices.
'the vector v must have the same size as A');
end;
if (~issparse(A))
A = sparse(A);
end;
% normalize the matrix
P = normout(A);
% options.approx_subiter: number of subiterations of power iterations
% [integer | {5}]
%
% Example:
% load cs-stanford;
% p = pagerank(A);
% and dt is the total time spent solving the system
%
% The matrix A should have the outlinks represented in the rows.
%
% This driver can compute PageRank using 4 different algorithms,
%
% pagerank(A,options) specifies optional parameters
% options.c: the teleportation coefficient [double | {0.85}]
% options.tol: the stopping tolerance [double | {1e-7}]
%
% Note 1: the 'approx' algorithm is the PageRank approximate personalized
% PageRank algorithm due to Gleich and Polito. It creates a set of
%
% [p flag hist dt] = pagerank(A)
%
% Compute the pagerank vector p for the directed graph A, with
% teleportation probability (1-c).
%
% flag is 1 if the method converged; hist returns the convergence history
% options.v: the personalization vector [vector | {uniform: 1/n}]
% options.maxiter maximum number of iterations [integer | {500}]
% options.verbose: extra output information [{0} | 1]
> pagerank-1.2(matlab).zip > pagerank.m, change:2006-06-30,size:22806b
function [x flag hist dt] = pagerank(A,optionsu)
% PAGERANK Compute the PageRank for a directed graph.
% p = pagerank(A,struct('alg','linsys',...
% 'linsys_solver',@(f,v,tol,its) gmres(f,v,[],tol, its)));
% pagerank(A,struct('alg','eval'));
% PageRank formulation.
%
% The output p satisfies p = c A'*D^{+} p + c d'*p v + (1-c) v and
% norm(p,1) = 1.
%
% The power method solves the eigensystem x = P''^T x.
'approx_subiter', 5);
if (nargin > 1)
options = merge_structs(optionsu, options);
end;
if (size(options.v) ~= size(A,1))
error('pagerank:invalidParameter', ...
%
[m n] = size(A);
if (m ~= n)
error('pagerank:invalidParameter', 'the matrix A must be square');
end;
options = struct('tol', 1e-7, 'maxiter', 500, 'v', ones(n,1)./n, ...
% p{1} is the vector computed from the 'power' algorithm
% p{2} is the vector computed from the 'gs' algorithm
% p{3} is the vector computed from the 'arnoldi' algorithm
% the default algorithm is the Arnoldi iteration for PageRank due to
% Grief and Golub. Other algorithms include gauss-seidel iterations,
% power iterations, a linear system formulation, or an approximate
% 'approx' | 'eval']
%
% options.linsys_solver: a function handle for the linear solver used
% with the linsys option [fh | {@(f,v,tol,its) bicgstab(f,v,tol,its)}]
% options.arnoldi_k: use a k dimensional arnoldi basis [intger | {8}]
% options.approx_bp: boundary probability to expand [float | 1e-3]
% options.approx_boundary: when to expand on the boundary [1 | {inf}]
%
% Revision 1.00
% 19 Octoboer 2005
%
%
% The driver does mainly parameter checking, then sends things off to one
% of the computational routines.
% active pages and runs until either norm(p(boundary),1) < options.bp or
% norm(p(boundary),inf) < options.bp, where the boundary is defined as