Ordinal Ranking for Google's PageRank

We present computationally efficient criteria that can guarantee correct ordinal ranking of Google's PageRank scores when they are computed with the power method (ordinal ranking of a list consists of assigning an ordinal number to each item in the list). We discuss the tightness of the ranking criteria, and illustrate their effectiveness for top k and bucket ranking. We present a careful implementation of the power method, combined with a roundoff error analysis that is valid for matrix dimensions $n<10^{14}$. To first order, the roundoff error depends neither on $n$ nor on the iteration count, but only on the maximal number of inlinks and the dangling nodes. The applicability of our ranking criterion is limited by the roundoff error from a single matrix vector multiply. Numerical experiments suggest that our criteria can effectively rank the top PageRank scores. We also discuss how to implement ranking for extremely large practical problems, by curbing roundoff error, reducing the matrix dimension, and using faster converging methods.