Perron vector optimization applied to search engines

• Published in: Applied numerical mathematics Vol. 75; pp. 77 - 99
• Main Author: Fercoq, Olivier
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.01.2014
• Subjects: Algorithms; Invariants; Mathematical analysis; Optimization; Perron–Frobenius theory; Ranking; Scalable algorithms; Search engines; Stochasticity; Strategy; Search engines; Scalable algorithms; Optimization; Perron–Frobenius theory
• ISSN: 0168-9274; 1873-5460
• DOI: 10.1016/j.apnum.2012.12.006

In the last years, Googleʼs PageRank optimization problems have been extensively studied. In that case, the ranking is given by the invariant measure of a stochastic matrix. In this paper, we consider the more general situation in which the ranking is determined by the Perron eigenvector of a nonnegative, but not necessarily stochastic, matrix, in order to cover Kleinbergʼs HITS algorithm. We also give some results for Tomlinʼs HOTS algorithm. The problem consists then in finding an optimal outlink strategy subject to design constraints and for a given search engine. We study the relaxed versions of these problems, which means that we should accept weighted hyperlinks. We provide an efficient algorithm for the computation of the matrix of partial derivatives of the criterion, that uses the low rank property of this matrix. We give a scalable algorithm that couples gradient and power iterations and gives a local minimum of the Perron vector optimization problem. We prove convergence by considering it as an approximate gradient method. We then show that optimal linkage strategies of HITS and HOTS optimization problems satisfy a threshold property. We report numerical results on fragments of the real web graph for these search engine optimization problems.