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Introducing Total Harmonic Resistance for Graph Robustness under Edge Deletions

Lukas Berner, Henning Meyerhenke

TL;DR

The paper tackles graph robustness under budgeted edge deletions ($k$-GRoDel) by introducing total harmonic resistance $R_h(G)$ as a robustness measure that remains meaningful on disconnected graphs. It compares $R_h(G)$ with the forest index $R_f(G)$ and shows THR tends to select more central edges rather than peripheral cuts, supported by exact small-graph analyses and a greedy algorithm adaptation. Empirical results on Berlin road networks and benchmark graphs indicate THR-based solutions yield more central edge sets, with higher centrality measures, albeit at higher computational cost than forest-index–based methods. Overall, THR provides a principled, practical robustness objective for edge-deletion attacks, with promising implications for protecting critical infrastructure networks.

Abstract

Assessing and improving the robustness of a graph $G$ are critical steps in network design and analysis. To this end, we consider the optimisation problem of removing $k$ edges from $G$ such that the resulting graph has minimal robustness, simulating attacks or failures. In this paper, we propose total harmonic resistance as a new robustness measure for this purpose - and compare it to the recently proposed forest index [Zhu et al., IEEE Trans.\ Inf.\ Forensics and Security, 2023]. Both measures are related to the established total effective resistance measure, but their advantage is that they can handle disconnected graphs. This is also important for originally connected graphs due to the removal of the $k$ edges. To compare our measure with the forest index, we first investigate exact solutions for small examples. The best $k$ edges to select when optimizing for the forest index lie at the periphery. Our proposed measure, in turn, prioritizes more central edges, which should be beneficial for most applications. Furthermore, we adapt a generic greedy algorithm to our optimization problem with the total harmonic resistance. With this algorithm, we perform a case study on the Berlin road network and also apply the algorithm to established benchmark graphs. The results are similar as for the small example graphs above and indicate the higher suitability of the new measure.

Introducing Total Harmonic Resistance for Graph Robustness under Edge Deletions

TL;DR

The paper tackles graph robustness under budgeted edge deletions (-GRoDel) by introducing total harmonic resistance as a robustness measure that remains meaningful on disconnected graphs. It compares with the forest index and shows THR tends to select more central edges rather than peripheral cuts, supported by exact small-graph analyses and a greedy algorithm adaptation. Empirical results on Berlin road networks and benchmark graphs indicate THR-based solutions yield more central edge sets, with higher centrality measures, albeit at higher computational cost than forest-index–based methods. Overall, THR provides a principled, practical robustness objective for edge-deletion attacks, with promising implications for protecting critical infrastructure networks.

Abstract

Assessing and improving the robustness of a graph are critical steps in network design and analysis. To this end, we consider the optimisation problem of removing edges from such that the resulting graph has minimal robustness, simulating attacks or failures. In this paper, we propose total harmonic resistance as a new robustness measure for this purpose - and compare it to the recently proposed forest index [Zhu et al., IEEE Trans.\ Inf.\ Forensics and Security, 2023]. Both measures are related to the established total effective resistance measure, but their advantage is that they can handle disconnected graphs. This is also important for originally connected graphs due to the removal of the edges. To compare our measure with the forest index, we first investigate exact solutions for small examples. The best edges to select when optimizing for the forest index lie at the periphery. Our proposed measure, in turn, prioritizes more central edges, which should be beneficial for most applications. Furthermore, we adapt a generic greedy algorithm to our optimization problem with the total harmonic resistance. With this algorithm, we perform a case study on the Berlin road network and also apply the algorithm to established benchmark graphs. The results are similar as for the small example graphs above and indicate the higher suitability of the new measure.
Paper Structure (21 sections, 3 theorems, 10 equations, 2 figures, 2 tables, 1 algorithm)

This paper contains 21 sections, 3 theorems, 10 equations, 2 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Contribution.
  3. Problem Statement and a new Robustness Measure
  4. Problem Statement and Notation
  5. Robustness Measures
  6. Related Work
  7. Comparison of Exact Solutions
  8. Discussion.
  9. Greedy Heuristic for $k$-GRoDel
  10. Total Harmonic Resistance loss after Deleting an Edge
  11. Forest Index loss after Deleting an Edge
  12. Edge Removal.
  13. Experimental Results
  14. Experimental Setup
  15. Case Study: Berlin Districts
  16. ...and 6 more sections

Key Result

Theorem 1

Given $G=(V,E)$, define the augmented Graph$G_* = (V_*, E_*)$ with a universal vertex $u^*$ which is connected to all other vertices: $V_* = V \cup \{u^*\}$ and $E_* = E \cup \{(v, u^*) : v\in V\}$. Then $d^{f}_{G}(u,v) = \mathbf{r}_{G_*}(u,v) ~ \forall u,v\in V$.

Figures (2)

  • Figure 1: Optimal solutions for $k=5$ on grid-like graphs using FI (left column) and THR (right column) as resistance measures. Edges highlighted in blue belong to the solution set.
  • Figure 2: Berlin case study result. Grey edges are the GreedyTHR solution; orange edges are the GreedyFI solution. The image is cropped -- not all edges in the solution are displayed. GreedyTHR finds four of seven river bridges while GreedyFI mostly finds residential roads. Image created using OpenStreetMap OpenStreetMap

Theorems & Definitions (7)

  • Theorem 1
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • proof
  • proof