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Rank monotonicity in centrality measures

Published online by Cambridge University Press:  26 July 2017

PAOLO BOLDI Open the ORCID record for PAOLO BOLDI [Opens in a new window] ,
ALESSANDRO LUONGO  and
SEBASTIANO VIGNA
PAOLO BOLDI
Affiliation:
Dipartimento di Informatica, Università degli Studi di Milano, Milano, Italy (e-mail: paolo.boldi@unimi.it, alessandro@luongo.pro, sebastiano.vigna@unimi.it)
ALESSANDRO LUONGO
Affiliation:
Dipartimento di Informatica, Università degli Studi di Milano, Milano, Italy (e-mail: paolo.boldi@unimi.it, alessandro@luongo.pro, sebastiano.vigna@unimi.it)
SEBASTIANO VIGNA
Affiliation:
Dipartimento di Informatica, Università degli Studi di Milano, Milano, Italy (e-mail: paolo.boldi@unimi.it, alessandro@luongo.pro, sebastiano.vigna@unimi.it)
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Abstract

A measure of centrality is rank monotone if after adding an arc x → y, all nodes with a score smaller than (or equal to) y have still a score smaller than (or equal to) y. If, in particular, all nodes with a score smaller than or equal to y get a score smaller than y (i.e., all ties with y are broken in favor of y), the measure is called strictly rank monotone. We prove that harmonic centrality is strictly rank monotone, whereas closeness is just rank monotone on strongly connected graphs, and that some other measures, including betweenness, are not rank monotone at all (sometimes not even on strongly connected graphs). Among spectral measures, damped scores such as Katz's index and PageRank are strictly rank monotone on all graphs, whereas the dominant eigenvector is strictly monotone on strongly connected graphs only.

Keywords

centralitynetwork analysisrank monotonicity

Information

Type
Research Article
Information
Network Science , Volume 5 , Issue 4 , December 2017 , pp. 529 - 550
Copyright
Copyright © Cambridge University Press 2017 

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