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.
PageRank
damping factor
Markov chains
1-19
Regular Paper
Paolo
Boldi
Paolo Boldi
Massimo
Santini
Massimo Santini
Sebastiano
Vigna
Sebastiano Vigna
10.4230/DagSemProc.07071.3
Creative Commons Attribution 4.0 International license
https://creativecommons.org/licenses/by/4.0/legalcode