Spectral Rank Monotonicity on Undirected Networks

TL;DR

This paper investigates how spectral ranking methods like PageRank behave when edges are added to undirected networks, revealing that PageRank is neither score nor rank monotone in such cases.

Contribution

The study demonstrates that spectral rankings such as PageRank do not maintain monotonicity properties in undirected networks, contrasting with their behavior in directed graphs.

Findings

PageRank is neither score nor rank monotone on undirected graphs.

Spectral ranking methods behave differently in undirected networks compared to directed ones.

Abstract

We study the problem of score and rank monotonicity for spectral ranking methods, such as eigenvector centrality and PageRank, in the case of undirected networks. Score monotonicity means that adding an edge increases the score at both ends of the edge. Rank monotonicity means that adding an edge improves the relative position of both ends of the edge with respect to the remaining nodes. It is known that common spectral rankings are both score and rank monotone on directed, strongly connected graphs. We show that, surprisingly, the situation is very different for undirected graphs, and in particular that PageRank is neither score nor rank monotone.