Recursive reordering and elimination method for efficient computation of PageRank problems

The PageRank model is widely utilized for analyzing a variety of scientific issues beyond its original application in modeling web search engines. In recent years, considerable research effort has focused on developing high-performance iterative methods to solve this model, particularly when the dimension is exceedingly large. However, due to the ever-increasing extent and size of data networks in various applications, the computational requirements of the PageRank model continue to grow. This has led to the development of new techniques that aim to reduce the computational complexity required for the solution. In this paper, we present a recursive 5-type lumping algorithm combined with a two-stage elimination strategy that leverage characteristics about the nonzero structure of the underlying network and the nonzero values of the PageRank coefficient matrix. This method reduces the initial PageRank problem to the solution of a remarkably smaller and sparser linear system. As a result, it leads to significant cost reductions for computing PageRank solutions, particularly in scenarios involving large and/or multiple damping factors. Numerical experiments conducted on over 50 real-world networks demonstrate that the proposed methods can effectively exploit characteristics of PageRank problems for efficient computations.