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A continuation method for computing the multilinear PageRank. (English) Zbl 07584140

Summary: The multilinear PageRank model [D. F. Gleich et al., SIAM J. Matrix Anal. Appl. 36, No. 4, 1507–1541 (2015; Zbl 1330.15029)] 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 nonunique. 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 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 \(\mathbb{R}^{n+1}\). 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.

MSC:

65F15 Numerical computation of eigenvalues and eigenvectors of matrices
15A69 Multilinear algebra, tensor calculus

Citations:

Zbl 1330.15029
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