document: 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 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 behavior of PageRank with respect to changes in α was discovered to be useful in link-spam detection ( 21 ) . 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 behavior 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 .
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