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.