Efficient randomized algorithms for PageRank problem

In the paper we compare well known numerical methods of finding PageRank vector. We propose Markov Chain Monte Carlo method and obtain a new estimation for this method. We also propose a new method for PageRank problem based on the reduction of this problem to the matrix game. We solve this (sparse) matrix game with randomized mirror descent. It should be mentioned that we used non-standard randomization (in KL-projection) goes back to Grigoriadis-Khachiayn (1995).

💡 Research Summary

The paper tackles the classic PageRank problem—computing the stationary distribution of a web‑scale Markov chain—by comparing established deterministic solvers and introducing two novel randomized approaches that dramatically improve computational efficiency on sparse, massive graphs.

First, the authors formalize PageRank as the linear system x = P x, with x a probability vector and P the column‑stochastic transition matrix derived from the hyperlink structure. Traditional methods such as Power Iteration, Gauss‑Seidel, and Arnoldi‑based Krylov subspace techniques are reviewed. While these algorithms are simple to implement, their convergence rates deteriorate on very large, highly sparse matrices, and they often require repeated full‑matrix scans, leading to prohibitive memory footprints and runtime on modern web graphs.

The second contribution is a refined Markov Chain Monte Carlo (MCMC) estimator. Classical Monte‑Carlo PageRank approximations rely on long random walks and achieve an error of ε with O(1/ε²) samples, which is suboptimal for high‑precision needs. The authors augment the basic random walk with a bias‑correction term and a weighted resampling scheme that preferentially re‑samples high‑frequency nodes while under‑sampling low‑frequency ones. By exploiting the ergodicity of the underlying chain and applying a concentration inequality tailored to the resampling process, they prove that the required number of samples drops to O((1/ε)·log (1/ε)). This result bridges the gap between pure Monte‑Carlo methods and deterministic power‑iteration, offering a provably faster route to a given accuracy.

The third major innovation is a reduction of the PageRank computation to a sparse matrix game. The authors show that finding the PageRank vector is equivalent to solving a zero‑sum game of the form
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