Optimal dismantling of directed networks

TL;DR

This work tackles the problem of dismantling directed networks by targeting their hierarchical, directional structure rather than treating them as undirected. It introduces trophic analysis-based NI centrality ($F_i$) and the TAD method, which removes nodes in descending $F_i$ to rapidly reduce the giant strongly connected component ($GSCC$) and induce large avalanche-like breakdowns. Across synthetic networks with varying degree heterogeneity and 15 real-world directed networks, TAD consistently outperforms state-of-the-art baselines, with especially strong performance in networks that preserve a moderate level of hierarchical structure. The study also provides mechanistic insights into which nodes sustain directed connectivity and how backward links shape global directionality, offering a principled, interpretable framework for designing more resilient directed systems.

Abstract

As a fundamental problem in network science, network dismantling focuses on identifying a set of critical nodes whose removal sharply reduces a network's connectivity and functionality. Potential applications include stopping rumor spread, blocking sentiment propagation, and controlling epidemics and pandemics. Previous studies have mainly focused on undirected networks, whereas many real-world networks are inherently directed, such as the World Wide Web and the global trade system. Moreover, the functionality of directed networks depends on the giant strongly connected component (GSCC), where nodes are mutually reachable. Considering both the directionality and heterogeneity of these networks, we propose a novel centrality measure, network incoherence (NI) centrality, and develop a trophic analysis-based dismantling (TAD) method, in which nodes are removed in descending order according to their NI centrality scores, aiming to efficiently dismantle directed networks by reducing the GSCC. When applied to a wide range of benchmark synthetic networks with varying degree heterogeneity and 15 real-world directed networks, our TAD method consistently outperforms existing state-of-the-art methods. Significantly, TAD also induces the largest maximum avalanches during the dismantling process, highlighting its ability to capture structurally critical nodes. These findings provide new insight into the structure-function relationship of directed networks and inform the design of more resilient systems against perturbations.

Figures (6)

• Figure 1: Illustration of the TAD process. (a) The network before and after node removal is shown. In the original network, node color (from dark red to light red) represents the attack order, while node size indicates importance. In the residual network, light gray nodes have been removed, and nodes within the purple region constitute the GSCC. (b–e) Overview of the TAD. The input is the adjacency matrix of the directed network (b). The evaluation of node importance involves three steps: calculating node trophic levels (c); determining the trophic level differences of links and constructing an adjusted undirected network (d); and computing the NI centrality score for each node (e). (f) Visualization of the dismantling process, where the GSCC size of the residual network decreases as nodes are sequentially removed in the order from dark to light red.
• Figure 2: Performance of TAD on SF networks with varying degree distribution exponents ($\lambda$ = 2.2, 2.5, 2.8, 3.2). (a) Comparison of TAD, MinSum, Finder, adaptive degree, and other methods on SF networks. The solid lines indicate the size of the GSCC as a function of the fraction of removed nodes. (b) The average Area Under the Curve (AUC) of the SF network over the entire dismantling process. The networks contain 1000 nodes and the error bars represent the standard deviations across 30 random realizations.
• Figure 3: The maximum avalanches caused by different methods. (a) The process of dismantling a directed network using various methods. The horizontal axis represents the fraction of nodes removed, and the vertical axis shows the size of the GSCC. The maximum avalanches are highlighted in thicker lines. (b)-(e) The size of the maximum avalanches caused by different methods across SF networks with different degree distribution exponent $\lambda$=2.2, 2.5, 2.8, and 3.2. The results are averaged over 30 realizations.
• Figure 4: The performance of different network dismantling methods across networks with varying $F$. The horizontal axis represents the trophic incoherence of a network, while the vertical axis corresponds to AUC, where a lower AUC indicates greater dismantling efficiency. (a) The results for ER networks ($\langle k \rangle=20$) generated with varying trophic incoherence using a simulated annealing method rodgers2022network. The results show that TAD achieves the lowest AUC when $F$ is below 0.62, indicating that it is the most effective dismantling method in networks with moderate hierarchical structure. (b) Results for SF networks ($\lambda = 2.8$) across different levels of trophic incoherence $F$. Since randomly generated SF networks rarely reach $F > 0.6$, we artificially increased $F$ by reversing a fraction of links. TAD exhibits the best performance (lowest AUC) when $F$ is small, confirming its effectiveness in networks with strong hierarchical structure. As $F$ increases beyond $0.67$, its AUC continues to rise and is eventually surpassed by other methods. This transition highlights the structural dependence of dismantling strategies and suggests that TAD is particularly suited to networks with moderate to low incoherence, where hierarchy plays a central role.
• Figure 5: AUC values of different dismantling methods applied to 15 real-world directed networks. The horizontal axis lists the networks from various domains (e.g., ecological, neural, scholarly, and social), and the vertical axis shows the average area under the GSCC curve (AUC) for each method. Lower AUC values indicate better dismantling performance. TAD consistently achieves the lowest AUC in all networks,demonstrating its superior effectiveness in fragmenting real-world directed networks. The final column shows the average AUC across all networks.