Multilevel Adaptive Aggregation for Markov Chains, with Application to Web Ranking

Abstract

Abstract. A multilevel adaptive aggregation method for calculating the stationary probability vector of an irreducible stochastic matrix is described. The method is a special case of the adaptive smooth aggregation and adaptive algebraic multigrid methods for sparse linear systems, and is also closely related to certain extensively studied iterative aggregation/disaggregation methods for Markov chains. In contrast to most existing approaches, our aggregation process does not employ any explicit advance knowledge of the topology of the Markov chain. Instead, adaptive agglomeration is proposed that is based on strength of connection in a scaled problem matrix, in which the columns of the original problem matrix at each recursive fine level are scaled with the current probability vector iterate at that level. Strength of connection is determined as in the algebraic multigrid method, and the aggregation process is fully adaptive, with optimized aggregates chosen in each step of the iteration and at all recursive levels. The multilevel method is applied to a set of stochastic matrices that provide models for web page ranking. Numerical tests serve to illustrate for which types of stochastic matrices the multilevel adaptive method may provide significant speedup compared to standard iterative methods. The tests also provide more insight into why Google's PageRank model is a successful model for determining a ranking of web pages.

An Accelerated Two-level Multigrid Method for Markov Chains

“…For computing numerical solutions of the linear system Ax b = with b the right-hand vector, certain multigrid methods have been presented in [24,28,29]. It is easy to find that the linear system (2) is in fact a special case of Ax b = when b = 0.…”

“…Here operators P and R are created by an automatic coarsening process described below. Then, starting from the description of multigrid methods for the linear system Ax b = [24,28,29], the twolevel multigrid method for Markov chains is proposed in Algorithm 1, where the matrix A is known in the linear system (2), G is an aggregation matrix generated by Algorithm 2, and iter denotes the number of cycles until the one-norm residual 1 …”

“…Here, in Algorithm 2, we adopt a strength-based aggregation procedure that proposed by De Sterck et al as our aggregation method, since it is able to improve an algebraically smooth error that varies slowly in a local neighborhood by scaling the original problem matrix A [24,25]. Otherwise go to step 2.…”

“…x a rather than the original coefficient matrix A (for details see [24]), where diag(·) denotes a diagonal matrix formed with the current iterate x k . Taking the similar way of defining the strength matrix S [24], then it follows that 1,if and max ( ), 0,otherwise,…”

“…For instance, the matrix splitting iterative methods [13,[15][16][17] Krylov, subspace methods [18][19][20][21] and some preconditioning techniques [6,9,17] and so on. What is more, based on the aggregation of Markov states, multigrid methods have been studied in the literature [22][23][24][25][26][27]. However, with the size of the Markov chains becomes large, the cost of multigrid methods is likely to have an increase.…”