PageRank as a Function of the Damping Factor

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 # that spreads uniformly part of the rank. The choice of # is eminently empirical, and in most cases the original suggestion # = 0.85 by Brin and Page is still used. Recently, however, the behaviour of PageRank with respect to changes in # was discovered to be useful in link-spam detection [21]. Moreover, an analytical justification of the value chosen for # is still missing. In this paper, we give the first mathematical analysis of PageRank when # changes. In particular, we show that, contrarily to popular belief, for real-world graphs values of # 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 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