PageRank as a function of the damping factor
Proceedings of the 14th International Conference on World Wide Web, page 557--566. Chiba, ACM, (2005)bibtex: Boldi2005PageRank.DOI: 10.1145/1060745.1060827
Abstract
PageRank is defined as the stationary state of a Markov chain. The chain is obtained by perturbing the transition matrix induced by a web graph with a damping factor α that spreads uniformly part of the rank. The choice of α is eminently empirical, and in most cases the original suggestion α = 0.85 by Brin and Page is still used. Recently, however, the behaviour of PageRank with respect to changes in α was discovered to be useful in link-spam detection21. Moreover, an analytical justification of the value chosen for α is still missing. In this paper, we give the first mathematical analysis of PageRank when α changes. In particular, we show that, contrarily to popular belief, for real-world graphs values of α close to 1 do not give a more meaningful ranking. Then, we give closed-form formulae for PageRank derivatives of any order, and an extension of the Power Method that approximates them with convergence O (tk αt) for the k-th derivative. Finally, we show a tight connection between iterated computation and analytical behaviour by proving that the k-th iteration of the Power Method gives exactly the PageRank value obtained using a Maclaurin polynomial of degree k. The latter result paves the way towards the application of analytical methods to the study of PageRank.
