Locally Optimal Percolation for Network Resilience Dismantling via Fiedler Vector Gradient Iterative Attack

TL;DR

This work tackles the disconnect between structural percolation and functional resilience by treating resilience as governed by the Laplacian's algebraic connectivity $λ_2$. It introduces the Fiedler Gradient Iterative Attack (FGIA), which uses the Fiedler vector gradient $|x_i - x_j|$ to rank edges and iteratively remove non-bridge edges while maintaining connectivity, achieving $O(n^3)$ time complexity. A key theoretical result is that the edge-induced change $δλ_2(e_{ij})$ is dominated by $(∇ε_{ij})^2$, tying edge importance to inter-community coupling via the gradient. Through hierarchical spectral bisection and bridge filtering, FGIA outperforms traditional structural attacks across synthetic and real networks, substantially degrading resilience with relatively few removals. This approach offers a universal, scalable spectral-gradient tool for controlled resilience disruption with potential applications in neuroscience and critical infrastructure protection.

Abstract

Network resilience, dynamically quantified by the Fiedler value (\(λ_2\),the second smallest eigenvalue of the Laplacian matrix) ensures functional stability and efficient energy transmission, yet also introduces vulnerabilities that dismantling the resilience of the network can cause a functional breakdown of the network. However, traditional percolation strategies focused on structural attacks often fail to effectively affect resilience and lack universal applicability. Here, we employ a Laplacian spectral perturbation approach to systematically identify and remove edges critical to resilience. We derive the sensitivity of \(λ_2\) to topological changes and employ the gradient of Fiedler vector to measure each edge's contribution of resilience, revealing an intrinsic relationship to community partition. Accordingly, we propose the Fiedler Gradient Iterative Attack (FGIA) algorithm, which constructs locally optimal edge removal sequences to maximize \(λ_2\) degradation with significantly lower computational cost than brute-force methods. Our results offer a rigorous approach for inducing controlled resilience collapse, with potential applications in neuroscience and critical infrastructure protection.

Figures (6)

• Figure 1: (a) The generation of edge-attack from an all- connected graph, with the red dashed line representing edges that have been randomly removed. (b) Time Series of dynamics given by Eq. \ref{['eq:dynamics']} in different graphs in (a), respectively. Black dashed lines mark the convergence time. (c) Convergence time $\tau_c$ versus the number of removed nodes. (d) The reciprocal of convergence time (Red dot line) and $\lambda_2$ (Green triangle line) versus the number of removed nodes.
• Figure 2: Status of each node $\psi_i$, $1 \leq i \leq 7$, versus time $t$ by, respectively, controlling thee typical edges based on gradient in the example network, where the edge color represents the gradient, respectively.
• Figure 3: Diagram of community partition and the Fiedler gradient. a demonstrative case in a Barabási–Albert (BA) network with $15$ nodes and attachment parameter $m=1$. The color bar shows the Fiedler gradient. The red dotted line represents the first partition, and the purple dotted lines represent the second partition.
• Figure 4: The performance of various dynamical edge-attack algorithms on resilience of synthetic networks is illustrated in the top row, quantified by the resident percentage of $\lambda_2$ (pRAC) as a function of the percentage of edge removal (pER). The results are based on synthetic network models comprising 30 nodes. The effectiveness of each attack strategy is further assessed using the area under the curve (AUC) of $\lambda_2$. (a) and (b) correspond to Barabási–Albert (BA) scale-free networks, (c) and (d) to Watts–Strogatz (WS) small-world networks, and (e) and (f) to Erdős–Rényi (ER) random networks. The inset in (f) provides an enlarged view of the region highlighted by the red dashed rectangle. The bottom row displays the corresponding AUC values for each method, aligned with the plots in the top row. The bar chart in (h-m) shows the AUC values obtained by different algorithms in the first row.
• Figure 5: The AUC values obtained from several representative attack algorithms across different synthetic network models. (a) In Barabási–Albert (BA) scale-free networks, AUC values are plotted against the network growth parameter $m$. (b) and (c) show the results for Watts–Strogatz (WS) small-world networks: in (b), AUC values vary with the rewiring probability $p_r$ (with average degree fixed at $k = 5$), and in (c), with $k$ (with fixed $p_r = 0.3$). (d) presents the results for Erdős–Rényi (ER) random networks, with AUC values shown as a function of the edge connection probability $p_e$. Different colors represent different attack algorithms, as indicated in the legend.