Updating pagerank with iterative aggregation after dating death
It should be noted that the direct method on the coarsest level, used at step 4 of Algorithm 3, is based on the following theorem..
As mentioned in Section 1, although the Power method is a stable and reliable  or even a fast iteration algorithm , accelerating the computation is still important, since every search engine crawls a huge number of pages, and each matrix multiplication in iteration is so expensive, requiring considerable resources both in terms of CPU and RAM, and hence, the rate of convergence deteriorates as the number of pages grows larger.
Isensee and Horton considered a kind of multilevel methods for the steady state solution of continuous-time and discrete-time Markov chains in [13, 14], respectively. proposed a multilevel adaptive aggregation method for calculating the stationary probability vector of Markov matrices in , as shown in their context, which is a special case of the adaptive smoothed aggregation  and adaptive algebraic multigrid methods  for sparse linear systems.
The central idea of multilevel aggregation method is to convert a large linear system to a smaller one by some aggregation strategies, and thus, the stationary state solution can be obtained in an efficient way.
Due to the large size of the web graph (over eight billion nodes ), computing Page Rank is faced with the big challenge of computational resources, both in terms of the space of CPU and RAM required and in terms of the speed of updating in time; that is, as a new crawl is completed, it can be soon available for searching.
Among all the numerical methods to compute Page Rank, the Power method is one of the standard ways for its stable and reliable performances , whereas the low rate of convergence is its fatal flaw.