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Monotonicity in undirected networks

Published online by Cambridge University Press:  02 February 2023

Paolo Boldi
Dipartimento di Informatica, Università degli Studi di Milano, Milan, Italy
Flavio Furia
Dipartimento di Informatica, Università degli Studi di Milano, Milan, Italy
Sebastiano Vigna*
Dipartimento di Informatica, Università degli Studi di Milano, Milan, Italy
*Corresponding author. Email:


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, improves 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.

Research Article
© The Author(s), 2023. Published by Cambridge University Press

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Action Editor: Ulrik Brandes

The authors have been supported by the FASTEN EU Project, H2020-ICT-2018-2020 (GA 825328).

A preliminary version of this paper appeared as Boldi, P., Furia, F., Vigna, S. (2022) as Spectral Rank Monotonicity on Undirected Networks in International Conference Complex Networks & Their Applications (pp. 234–246). Cham: Springer.


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