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In the study of the behavior of centrality measures with respect to networkmodifications, score monotonicity means that adding an arc increases thecentrality score of the target of the arc; rank monotonicity means that addingan arc improves the importance of the target of the arc relative to theremaining nodes. It is known that score and rank monotonicity hold in directedgraphs for almost all centrality measures. In undirected graphs one expectsthat the corresponding properties (where both endpoints of the new edge enjoythe increase in score/rank) hold when adding a new edge. However, recentresults have shown that in undirected networks this is not true: for manycentrality measures, it is possible to find situations where adding an edgereduces the rank of one of its two endpoints. In this paper we introduce aweaker condition for undirected networks, semi-monotonicity, in which just oneof the endpoints of a new edge is required to enjoy score or rank monotonicity.We show that this condition is satisfied by closeness and betweennesscentrality, and that harmonic centrality satisfies it in an even strongersense.

Score and Rank Semi-Monotonicity for Closeness, Betweenness and Harmonic Centrality