Constructions of betweenness-uniform graphs from trees

TL;DR

This paper investigates a graph construction method aimed at creating graphs with uniform betweenness centrality, disproves its universality for trees with diameter at least three, and highlights open problems in characterizing such graphs.

Contribution

It demonstrates that the blow-up construction cannot produce betweenness-uniform graphs from certain trees, challenging previous assumptions about its universality.

Findings

The blow-up construction fails for trees with diameter ≥ 3.

Disproved the conjecture about the universality of the construction.

Open problem: characterizing graphs that can be generated as betweenness-uniform by this method.

Abstract

Betweenness centrality is a measure of the importance of a vertex x inside a network based on the fraction of shortest paths passing through x. We study a blow-up construction that has been shown to produce graphs with uniform distribution of betweenness. We disprove the conjecture about this procedure's universality by showing that trees with a diameter at least three cannot be transformed into betweenness-uniform by the blow-up construction. It remains open to characterize graphs for which the blow-up construction can produce betweenness-uniform graphs.