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 $\tilde{G}$ be the graph obtained by perturbing the weight of a single edge $[u,v]\in E$ by an amount $\Delta w_{uv}$, so that its new weight becomes $1+\Delta w_{uv}$, 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 $\tilde{G}$, using matrix perturbation theory and the Woodbury identity. We derive closed-form expressions for the perturbed resistance distance ${\tilde{r}}_{ij}$ and perturbed Kirchhoff index $\mathrm{Kf}\left(\tilde{G}\right)$, providing analytical insight into the global impact of local structural changes. Through extremal analysis on complete and path graphs, we establish tight bounds for $\mathrm{Kf}\left(\tilde{G}\right)$ 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.