Uniform Edge Betweenness Centrality

TL;DR

This paper explores graphs with uniform edge betweenness centrality, revealing their rarity among small graphs and presenting methods to construct infinite classes of such graphs.

Contribution

It identifies the rarity of non-edge-transitive graphs with uniform edge betweenness centrality and introduces methods to generate infinite classes of these graphs.

Findings

Only four non-edge-transitive graphs with uniform edge betweenness centrality among 11.9 million small graphs

Edge-transitivity guarantees uniform edge betweenness centrality, but is not necessary

Methods are provided to construct infinite classes of graphs with this property

Abstract

The edge betweenness centrality of an edge is loosely defined as the fraction of shortest paths between all pairs of vertices passing through that edge. In this paper, we investigate graphs where the edge betweenness centrality of edges is uniform. It is clear that if a graph G is edge-transitive (its automorphism group acts transitively on its edges) then G has uniform edge betweenness centrality. However this sufficient condition is not necessary. Graphs that are not edge-transitive but have uniform edge betweenness centrality appear to be very rare. Of the over 11.9 million connected graphs on up to ten vertices, there are only four graphs that are not edge-transitive but have uniform edge betweenness centrality. Despite this rarity among small graphs, we present methods for creating infinite classes of graphs with this unusual combination of properties.