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Analytical study of resistance distance and Kirchhoff index under edge perturbations in weighted graphs

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  • Sardar, Muhammad Shoaib

Abstract

Let G=(V,E) be a connected, undirected graph with unit edge weights. Let G̃ be the graph obtained by perturbing the weight of a single edge [u,v]∈E by an amount Δwuv, so that its new weight becomes 1+Δwuv, while all other edge weights remain unchanged. In this paper, we analyze the effect of such localized edge perturbations on the resistance distance and Kirchhoff index of G̃, using matrix perturbation theory and the Woodbury identity. We derive closed-form expressions for the perturbed resistance distance r̃ij and perturbed Kirchhoff index Kf(G̃), providing analytical insight into the global impact of local structural changes. Through extremal analysis on complete and path graphs, we establish tight bounds for Kf(G̃) under single-edge perturbations and derive first-order sensitivity approximations to quantify the influence of individual edge weights. Building on this foundation, we propose the Max-Kirchhoff Impact Edge Detection (MKIED) technique to locate edges that have the greatest influence on the Kirchhoff index. Experiments on real-world networks, including Facebook, the Karate Club, and Les Miserables, illustrate the method’s efficacy in identifying structurally significant edges, frequently correlating to bridges or hub-hub connections. The results underscore the Kirchhoff index as an effective instrument for assessing structural vulnerability in complex networks.

Suggested Citation

  • Sardar, Muhammad Shoaib, 2025. "Analytical study of resistance distance and Kirchhoff index under edge perturbations in weighted graphs," Chaos, Solitons & Fractals, Elsevier, vol. 199(P3).
    • Handle: RePEc:eee:chsofr:v:199:y:2025:i:p3:s0960077925009105
    DOI: 10.1016/j.chaos.2025.116897
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    References listed on IDEAS

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    1. Jiang, Zhuozhuo & Yan, Weigen, 2017. "Resistance between two nodes of a ring network," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 484(C), pages 21-26.
    2. R. B. Bapat & Somit Gupta, 2010. "Resistance distance in wheels and fans," Indian Journal of Pure and Applied Mathematics, Springer, vol. 41(1), pages 1-13, February.
    3. Sardar, Muhammad Shoaib & Pan, Xiang-Feng & Xu, Si-Ao, 2020. "Computation of resistance distance and Kirchhoff index of the two classes of silicate networks," Applied Mathematics and Computation, Elsevier, vol. 381(C).
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    5. Sardar, Muhammad Shoaib & Pan, Xiang-Feng & Xu, Shou-Jun, 2024. "Computation of the resistance distance and the Kirchhoff index for the two types of claw-free cubic graphs," Applied Mathematics and Computation, Elsevier, vol. 473(C).
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