On the initial value of PageRank

TL;DR

This paper explores how initial vertex values influence PageRank in undirected networks, demonstrating that specific initializations lead to PageRank proportional to vertex degrees, supported by analytical, simulated, and empirical evidence.

Contribution

It analytically and empirically shows the dependence of PageRank on initial values in undirected networks, a less-studied aspect of PageRank behavior.

Findings

PageRank depends on initial vertex values in undirected networks.

When initial values are proportional to degrees, PageRank becomes proportional to degrees.

Initial values significantly affect PageRank localization.

Abstract

Google employs PageRank to rank web pages, determining the order in which search results are presented to users based on their queries. PageRank is primarily utilized for directed networks, although there are instances where it is also applied to undirected networks. In this paper, we have applied PageRank to undirected networks, showing that a vertex's PageRank relies on its initial value, often referred to as an intrinsic, non-network contribution. We have analytically proved that when the initial value of vertices is either proportional to their degrees or set to zero, the PageRank values of the vertices become directly proportional to their degrees. Simulated and empirical data are employed to bolster our research findings. Additionally, we have investigated the impact of initial values on PageRank localization.