What does it mean to converge in rank?

We give the first rigourous, formal definition of convergence in rank of an iterative ranking algorithm - an intuitive notion whose formalization is more challenging than it might at first appear. We then compare, in the context of the well known PageRank algorithm, the rate of convergence in rank with the “classic” rate of convergence of the score vector. Even though both might appear to depend essentially on the same factors, subtle differences make it possible for either to be arbitrarily slow even when the other is quite fast. Thus, making predictions on one based on the other can be completely misleading.