  In the study of the behavior of centrality measures with respect to network
modifications, 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 relative to the
remaining nodes. It is known that score and rank monotonicity hold in directed
graphs for almost all centrality measures. In undirected graphs one expects
that the corresponding properties (where both endpoints of the new edge enjoy
the increase in score/rank) hold when adding a new edge. However, recent
results have shown that in undirected networks this is not true: for many
centrality measures, it is possible to find situations where adding an edge
reduces the rank of one of its two endpoints. In this paper we introduce a
weaker condition for undirected networks, semi-monotonicity, in which just one
of the endpoints of a new edge is required to enjoy score or rank monotonicity.
We show that this condition is satisfied by closeness and betweenness
centrality, and that harmonic centrality satisfies it in an even stronger
sense.
