A continuation method for computing the multilinear PageRank. Issue 4 (24th January 2022)

• Record Type: Journal Article
• Authors: Bucci, Alberto; Poloni, Federico
• Is Part Of: Numerical linear algebra with applications. Volume 29:Issue 4(2022)
• Page Start: n/a
• Page End: n/a
• Publication Date: 2022-01-24
• Subjects: higher‐order Markov chain -- nonnegative tensor -- numerical continuation -- PageRank -- tensor eigenvalue; Algebras, Linear -- Periodicals; 512.5
• DOI: 10.1002/nla.2432
• Languages: English
• ISSNs: 1070-5325

Abstract

Abstract: The multilinear PageRank model [Gleich et al., SIAM J Matrix Anal Appl, 2015;36(4):1507–41] is a tensor‐based generalization of the PageRank model. Its computation requires solving a system of polynomial equations that contains a parameter α ∈ [ 0, 1 ) . For α ≈ 1, this computation remains a challenging problem, especially since the solution may be nonunique. Extrapolation strategies that start from smaller values of α 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 α . In this article, 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 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.