A Tale of Two Limits: An Extremal Pagerank Problem

TL;DR

This paper investigates how the Pagerank vector can vary significantly with the jumping parameter on a directed graph, revealing a maximum discrepancy of approximately 1.16, especially near the limit as the parameter approaches 1.

Contribution

It demonstrates the potential for large differences in Pagerank vectors due to the jumping parameter, providing a specific construction that achieves near-maximal discrepancy.

Findings

Maximum discrepancy of about 1.16 between Pagerank vectors.

Discrepancy occurs when the jumping parameter is close to 1.

The discrepancy can be as large as \\sqrt{67/50} for certain graphs.

Abstract

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.