# Perron-based algorithms for the multilinear pagerank

## TL;DR

This paper introduces new Perron-based algorithms for solving the multilinear PageRank problem, improving reliability and ability to handle more complex instances by leveraging quadratic vector equation theory and homotopy continuation.

## Contribution

It develops novel fixed-point algorithms and a homotopy strategy for multilinear PageRank, addressing solution existence and computational challenges.

## Key findings

- Proves existence of a minimal solution distinct from the stochastic one.
- Develops more reliable algorithms capable of solving larger problem sets.
- Demonstrates improved performance over existing methods.

## Abstract

We consider the multilinear pagerank problem studied in [Gleich, Lim and Yu, Multilinear Pagerank, 2015], 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 homotopy continuation strategy. The resulting numerical method is more reliable than the existing alternatives, being able to solve a larger number of problems.