Monotonicity in Undirected Networks

TL;DR

This paper investigates how adding new edges in undirected networks affects various centrality measures, revealing that many classical centralities are not monotone and can decrease in importance when new relationships are formed.

Contribution

The study demonstrates that many well-known centrality measures are not monotone in undirected networks, contrasting with directed cases, and highlights the non-beneficial effects of new edges on node importance.

Findings

Most classical centralities are not rank monotone in undirected networks.

Betweenness and PageRank are not score monotone in undirected networks.

Adding edges can decrease the importance of nodes according to several centrality measures.

Abstract

Is it always beneficial to create a new relationship (have a new follower/friend) in a social network? This question can be formally stated as a property of the centrality measure that defines the importance of the actors of the network. Score monotonicity means that adding an arc increases the centrality score of the target of the arc; rank monotonicity means that adding an arc improves the importance of the target of the arc relatively to the remaining nodes. It is known that most centralities are both score and rank monotone on directed, strongly connected graphs. In this paper, we study the problem of score and rank monotonicity for classical centrality measures in the case of undirected networks: in this case, we require that score, or relative importance, improve at both endpoints of the new edge. We show that, surprisingly, the situation in the undirected case is very different, and in particular that closeness, harmonic centrality, betweenness, eigenvector centrality, Seeley's index, Katz's index, and PageRank are not rank monotone; betweenness and PageRank are not even score monotone. In other words, while it is always a good thing to get a new follower, it is not always beneficial to get a new friend.