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

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.