A continuation method for computing the multilinear Pagerank

TL;DR

This paper introduces a predictor-corrector continuation algorithm for computing multilinear PageRank, addressing challenges near $oldsymbol{ ext{α} o 1}$ and improving reliability over existing methods.

Contribution

It develops a novel continuation method that ensures stable computation of multilinear PageRank solutions across the entire parameter range.

Findings

The proposed method outperforms existing strategies in numerical experiments.

Global properties of the solution curve are established to guarantee algorithm stability.

The approach effectively handles non-uniqueness issues near $oldsymbol{ ext{α} o 1}$.

Abstract

The multilinear Pagerank model [Gleich, Lim and Yu, 2015] is a tensor-based generalization of the Pagerank model. Its computation requires solving a system of polynomial equations that contains a parameter $\alpha \in [0,1)$. For $\alpha \approx 1$, this computation remains a challenging problem, especially since the solution may be non-unique. Extrapolation strategies that start from smaller values of $\alpha$ and `follow' the solution by slowly increasing this parameter have been suggested; however, there are known cases where these strategies fail, because a globally continuous solution curve cannot be defined as a function of $\alpha$. In this paper, we improve on this idea, by employing a predictor-corrector continuation algorithm based on a more general representation of the solutions as a curve in $\mathbb{R}^{n+1}$. We prove several global properties of this curve that ensure the good behavior of the algorithm, and we show in our numerical experiments that this method is significantly more reliable than the existing alternatives.