A Deeper Investigation of 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 $alpha$ that spreads uniformly part of the rank. The choice of $alpha$ is eminently empirical, and in most cases the original suggestion $alpha=0.85$ by Brin and Page is still used. In this paper, we give a mathematical analysis of PageRank when $alpha$ changes. In particular, we show that, contrarily to popular belief, for real-world graphs values of $alpha$ close to $1$ do not give a more meaningful ranking. Then, we give closed-form formulae for PageRank derivatives of any order, and by proving that the $k$-th iteration of the Power Method gives exactly the PageRank value obtained using a Maclaurin polynomial of degree $k$, we show how to obtain an approximation of the derivatives. Finally, we view PageRank as a linear operator acting on the preference vector and show a tight connection between iterated computation and derivation.