Introducing Total Harmonic Resistance for Graph Robustness under Edge Deletions

TL;DR

This paper introduces total harmonic resistance as a new robustness measure for graphs, especially under edge deletions, and demonstrates its advantages over existing measures through theoretical analysis and practical case studies.

Contribution

The paper proposes total harmonic resistance as a novel robustness measure that handles disconnected graphs and prioritizes central edges, improving upon existing measures like the forest index.

Findings

Total harmonic resistance prioritizes central edges for robustness.

The new measure performs better in case studies on real and benchmark graphs.

The greedy algorithm effectively optimizes the robustness measure.

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.