Perron-based algorithms for the multilinear PageRank

Abstract

We consider the multilinear PageRank problem, studied in a 2015 paper by Gleich, Lim and Yu, which is a system of quadratic equations with stochasticity and nonnegativity constraints. We use the theory of quadratic vector equations to prove several properties of its solutions and suggest new numerical algorithms. In particular, we prove the existence of a certain minimal solution, which does not always coincide with the stochastic one that is required by the problem. We use an interpretation of the solution as a Perron eigenvector to devise new fixed-point algorithms for its computation and pair them with a continuation strategy based on a perturbative approach. The resulting numerical method is more reliable than the existing alternatives, being able to solve a larger number of problem