A Note on the PageRank of Undirected Graphs

TL;DR

This paper investigates the relationship between PageRank and degree distribution in undirected graphs, proving proportionality for specific cases and providing bounds for the general case.

Contribution

It proves proportionality of PageRank to degree distribution for a specific personalization vector and establishes bounds for the difference in general cases.

Findings

PageRank is proportional to degree for certain personalization vectors.

Provides upper and lower bounds for the difference between PageRank and degree distribution.

Shows that in general, PageRank is not exactly proportional to degree in undirected graphs.

Abstract

The PageRank is a widely used scoring function of networks in general and of the World Wide Web graph in particular. The PageRank is defined for directed graphs, but in some special cases applications for undirected graphs occur. In the literature it is widely noted that the PageRank for undirected graphs are proportional to the degrees of the vertices of the graph. We prove that statement for a particular personalization vector in the definition of the PageRank, and we also show that in general, the PageRank of an undirected graph is not exactly proportional to the degree distribution of the graph: our main theorem gives an upper and a lower bound to the L_1 norm of the difference of the PageRank and the degree distribution vectors.