  We uncover a new relation between Closeness centrality and the Condorcet
principle. We define a Condorcet winner in a graph as a node that compared to
any other node is closer to more nodes. In other words, if we assume that nodes
vote on a closer candidate, a Condorcet winner would win a two-candidate
election against any other node in a plurality vote. We show that Closeness
centrality and its random-walk version, Random-Walk Closeness centrality, are
the only classic centrality measures that are Condorcet consistent on trees,
i.e., if a Condorcet winner exists, they rank it first. While they are not
Condorcet consistent in general graphs, we show that Closeness centrality
satisfies the Condorcet Comparison property that states that out of two
adjacent nodes, the one preferred by more nodes has higher centrality. We show
that Closeness centrality is the only regular distance-based centrality with
such a property.
