A Tale of Two Limits: An Extremal Pagerank Problem

For a directed graph, the Pagerank algorithm emulates a random walker on the graph that occasionally "jumps" to a random vertex based on a jumping parameter $\alpha$. Upon completion, the algorithm generates a stochastic vector whose entries correspond to the limiting probability that the walker will be at that vertex. This vector is a right eigenvector of a corresponding Markov trasition matrix. Undoubtedly, this vector can drastically change based upon the jumping parameter $\alpha$. In this article, we investigate the maximum possible discrepancy for different Pagerank vectors on the same unweighted directed (perhaps with loops) graph as measured by the 2-norm. We show that the limsup of this discrepancy can be as large as $\sqrt{\frac{67}{50}}$ using a very specific construction. (For contrast, the norm of the difference for any two stochastic vectors is at most $\sqrt{2}$.) Interestingly, on this construction this discrepancy occurs when $\alpha = 1$ and when $\alpha$ is very close to 1.