Accelerated Evolving Set Processes for Local PageRank Computation

Abstract

This work proposes a novel framework based on nested evolving set processes to accelerate Personalized PageRank (PPR) computation. At each stage of the process, we employ a localized inexact proximal point iteration to solve a simplified linear system. We show that the time complexity of such localized methods is upper bounded by $\min \{\tilde{\mathcal{O}}(R^2/\hat{I}{\mu }^2),\tilde{\mathcal{O}}(m)\}$ to obtain an $\hat{I}\mu$-approximation of the PPR vector, where $m$ denotes the number of edges in the graph and $R$ is a constant defined via nested evolving set processes. Furthermore, the algorithms induced by our framework require solving only $\tilde{\mathcal{O}}(1/\sqrt{\hat{I}}\pm )$ such linear systems, where $\hat{I}\pm$ is the damping factor. When $1/\hat{I}{\mu }^2\ll m$, this implies the existence of an algorithm that computes an $epsilon$-approximation of the PPR vector with an overall time complexity of $\tilde{\mathcal{O}}\left(R^2/(\sqrt{\hat{I}}\pm \hat{I}{\mu }^2)\right)$, independent of the underlying graph size. Our result resolves an open conjecture from existing literature. Experimental results on real-world graphs validate the efficiency of our methods, demonstrating significant convergence in the early stages.