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Cut-edge centralities in an undirected graph

Dario Bini, Steve Kirkland, Guy Latouche, Beatrice Meini

TL;DR

This work revisits the cut-edge centrality defined via Kemeny’s constant, proposing a numerically stable, regularization-free computation $c(e)$ that remains well-defined for bridges. It provides explicit κ-based expressions for one-path graphs and trees with three branches, extends the framework to graphs with loops, and offers a stochastic-complement interpretation that clarifies the underlying random-walk dynamics. The authors validate the approach with synthetic experiments and a real road-network (Pisa), demonstrating numerical stability and the expected tendency for central cut-edges to connect subgraphs of similar size. Overall, the paper delivers both theoretical insights and practical tools for robust edge centrality analysis in undirected graphs.

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.

Cut-edge centralities in an undirected graph

TL;DR

This work revisits the cut-edge centrality defined via Kemeny’s constant, proposing a numerically stable, regularization-free computation that remains well-defined for bridges. It provides explicit κ-based expressions for one-path graphs and trees with three branches, extends the framework to graphs with loops, and offers a stochastic-complement interpretation that clarifies the underlying random-walk dynamics. The authors validate the approach with synthetic experiments and a real road-network (Pisa), demonstrating numerical stability and the expected tendency for central cut-edges to connect subgraphs of similar size. Overall, the paper delivers both theoretical insights and practical tools for robust edge centrality analysis in undirected graphs.

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.

Paper Structure

This paper contains 12 sections, 16 theorems, 56 equations, 12 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Kemeny's constant
  4. Kemeny's constant of tree graphs with loops
  5. Centrality of cut-edges
  6. Centrality of edges in one-path graphs
  7. Centrality of edges in trees formed by three branches
  8. A generalization
  9. Some numerical experiments
  10. Synthetic data
  11. The case of a road map
  12. Conclusions

Key Result

Theorem 1

Suppose that $G$ is a connected, undirected graph on $n$ vertices, without loops, with degree sequence $d_1 , \ldots, d_n$. For each $j, k = 1, \ldots, n$ with $j\ne k$, let $\sigma_{j,k}$ denote the number of spanning forests consisting of two trees, one of which contains vertex $j$ and the other o

Figures (12)

  • Figure 1: On the left, the star graph of Example \ref{['ex1']} for $n=8$. On the right, the graph of Example \ref{['ex2']} with $n=6$.
  • Figure 2: Class of graphs formed by $n$ vertices. Vertex $i$ is connected by an edge to vertex $i+1$, for $i=1,\ldots,n-1$; vertices 1 and $n$ may have a loop. Removing the edge $(m,m+1)$ with the methodology of abcmp yields two disjoint graphs that belong to the same class.
  • Figure 3: Graph of kind $E_{p,q,r}$ formed by 3 branches.
  • Figure 4: Plot of the function $c(i)$ for $p=100$ and for different values of $q$ and $r$ with $i$ in the range $[1, p]$.
  • Figure 5: Tree graph formed by four branches.
  • ...and 7 more figures

Theorems & Definitions (30)

  • Theorem 1
  • Corollary 1
  • Remark 1
  • Theorem 2
  • proof
  • Corollary 2
  • Remark 2
  • Proposition 1
  • proof
  • Theorem 3
  • ...and 20 more