Cut-edge centralities in an undirected graph

TL;DR

This paper revisits a centrality measure for cut-edges in undirected graphs, providing more stable computation methods and explicit formulas for specific graph classes, confirming its practical effectiveness.

Contribution

It introduces a numerically stable expression for the cut-edge centrality measure and derives explicit formulas for certain graph classes, enhancing computational reliability.

Findings

The new expression improves numerical stability.

Explicit formulas are derived for one-path graphs and trees.

Numerical tests confirm the measure's good physical behavior.

Abstract

A centrality measure of the cut-edges of an undirected graph, given in [Altafini et al.~SIMAX 2023] and based on Kemeny's constant, is revisited. A numerically more stable expression is given to compute this measure, and an explicit expression is provided for some classes of graphs, including one-path graphs and trees formed by three or more branches. These results theoretically confirm the good physical behaviour of this centrality measure, experimentally observed in [Altafini et al.~SIMAX 2023]. Numerical tests are reported to check the stability and to confirm the good physical behaviour.