  We perform the first axiomatic analysis of medial centrality measures. These
measures, also called betweenness-like centralities, assess the role of a node
in connecting others in the network. We focus on a setting with one target node
and several source nodes. We consider three classic medial centrality measures
adapted to this setting: Betweenness Centrality, Stress Centrality and Random
Walk Betweenness Centrality. While Betweenness and Stress Centralities assume
that the information in the network follows shortest paths, Random Walk
Betweenness Centrality assumes it moves randomly along the edges. We develop
the first axiomatic characterizations of all three measures. Our analysis shows
that Random Walk Betweenness, while conceptually different, shares several
common properties with classic Betweenness and Stress Centralities.
