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

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 ...

• Type: Journal
• Year: 2008
• Where: SIAMSC
• Authors: Hans De Sterck, Thomas A. Manteuffel, Stephen F. McCormick, Quoc Nguyen, John Ruge
• Keywords: Adaptive, Algebraic Multigrid Method, Markov Chains, SIAMSC 2008