Parallelization techniques for accelerating PageRank computation

PageRank is a probability distribution used to represent the likelihood that a person randomly clicking on links will arrive at any particular page. Let G = [gij ]i,j=1 be a Web graph adjacency matrix with elements gij = 1 when there is a link from page j to page i, with i 6= j, and zero otherwise. From this matrix we can construct a transition matrix P = [pij ] n i,j=1 as follows: pij = gij cj if cj 6= 0 and 0 otherwise, where cj = ∑n i=1 gij , 1 ≤ j ≤ n, represents the number of out-links. For pages with a nonzero number of out-links the matrix P is column stochastic. In this case, the PageRank vector can be obtained by solving Pπ = π. The Power method is one of the oldest and simplest iterative methods for solving this eigenvector problem. When the matrix P ≥ O is irreducible and stochastic, the Power method converges to the eigenvector corresponding to λmax = 1. However, the Web contains many pages without out-links. In this case, the matrix P is non-stochastic and the Power method can not be used. Moreover, the matrix irreducibility is not satisfied for a Web graph. In order to overcome these difficulties, Page and Brin [2] change the transition matrix P to a column stochastic matrix P̄ = α(P + vd ) + (1− α)ve , where d ∈ < is defined by di = 1 if and only if ci = 0 and the vector v ∈ < is some probability distribution over pages. Then, setting α such that 0 < α < 1, the Power method can be used to solve the stationary distribution of the ergodic Markov chain defined by P̄ π = π.