Closeness Centrality via the Condorcet Principle

TL;DR

This paper explores the relationship between Closeness centrality and the Condorcet principle, establishing conditions under which Closeness centrality uniquely satisfies Condorcet consistency and related properties in graphs.

Contribution

It introduces the concept of Condorcet winners in graphs and proves that Closeness centrality uniquely satisfies Condorcet consistency on trees and a related comparison property.

Findings

Closeness centrality and Random-Walk Closeness are Condorcet consistent on trees.

Closeness centrality satisfies the Condorcet Comparison property in general graphs.

Closeness centrality is the only regular distance-based centrality with the Condorcet Comparison property.

Abstract

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.