Abstract
AbstractThe 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 . For , this computation remains a challenging problem, especially since the solution may be nonunique. Extrapolation strategies that start from smaller values of 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 . 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 . 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.
Highlights
Set x⊗m = x ⊗ x ⊗ · · · ⊗ x, m times where ⊗ denotes the Kronecker product [7, Section 11.4], and e = [1, 1 . . . , 1]T ∈ Rn, the vector of all ones
The Multilinear Pagerank problem consists in finding a stochastic solution x ∈ Rn to the equation x = fα(x), fα(x) = αR(x⊗m) + (1 − α)v, (1)
We give a theoretical contribution, proving that there is a connected solution curve that allows us to track the solution correctly up to α = 1 in problems without singular points, and we present a practical algorithm that allows us to solve multilinear Pagerank problem with higher reliability than the existing methods
Summary
Sufficient conditions for the uniqueness of solutions have been proposed in literature [4, 9]; the simplest (but weakest) of them is α In view of this property, various of the algorithms proposed fall in the setting of extrapolation methods, where the a sequence of solutions x(1), x(2), x(3),. H(x, α) := fα(x) − x = αR(x⊗m) + (1 − α)v − x, H : Rn × R → Rn as a curve in Rn+1, and use extrapolation techniques in the family of continuation algorithms [1] to compute points following this curve. The algorithms we use belong to the family of predictor-corrector continuation algorithms In cases such as the one, the curve makes an S-bend, and we cannot consider it anymore (locally) as a parametrized curve with the coordinate α as a parameter: instead, we compute points following the curve, using.
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