Rank monotonicity in centrality measures

Abstract

AbstractA measure of centrality isrank monotoneif after adding an arcx→y, all nodes with a score smaller than (or equal to)yhave still a score smaller than (or equal to)y. If, in particular, all nodes with a score smaller than or equal toyget a score smaller thany(i.e., all ties withyare broken in favor ofy), the measure is calledstrictly 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,dampedscores 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.