A Deeper Investigation of PageRank as a Function of the Damping Factor

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 $alpha$ 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.