PageRank and rank-reversal dependence on the damping factor

TL;DR

This paper investigates how the PageRank algorithm's stability depends on the damping factor, revealing frequent rank-reversals across a broad range of values and identifying more stable configurations.

Contribution

It provides a detailed analysis of rank-reversal phenomena in PageRank as a function of the damping factor, including new insights into network structures affecting stability.

Findings

Rank-reversal occurs frequently over a broad range of damping factors.

Correlation of PageRank vectors drops rapidly as damping factor deviates from 0.85.

Relative rank stability is higher around a damping factor of 0.65.

Abstract

PageRank (PR) is an algorithm originally developed by Google to evaluate the importance of web pages. Considering how deeply rooted Google's PR algorithm is to gathering relevant information or to the success of modern businesses, the question of rank-stability and choice of the damping factor (a parameter in the algorithm) is clearly important. We investigate PR as a function of the damping factor d on a network obtained from a domain of the World Wide Web, finding that rank-reversal happens frequently over a broad range of PR (and of d). We use three different correlation measures, Pearson, Spearman, and Kendall, to study rank-reversal as d changes, and show that the correlation of PR vectors drops rapidly as d changes from its frequently cited value, $d_0=0.85$. Rank-reversal is also observed by measuring the Spearman and Kendall rank correlation, which evaluate relative ranks rather than absolute PR. Rank-reversal happens not only in directed networks containing rank-sinks but also in a single strongly connected component, which by definition does not contain any sinks. We relate rank-reversals to rank-pockets and bottlenecks in the directed network structure. For the network studied, the relative rank is more stable by our measures around $d=0.65$ than at $d=d_0$.