Finding an optimal subset of nodes in a network that is able to efficiently disrupt the functioning of a corrupt or criminal organization or contain an epidemic or the spread of misinformation is a highly relevant problem of network science. In this paper, we address the generalized network-dismantling problem, which aims at finding a set of nodes whose removal from the network results in the fragmentation of the network into subcritical network components at minimal overall cost. Compared with previous formulations, we allow the costs of node removals to take arbitrary nonnegative real values, which may depend on topological properties such as node centrality or on nontopological features such as the price or protection level of a node. Interestingly, we show that nonunit costs imply a significantly different dismantling strategy. To solve this optimization problem, we propose a method which is based on the spectral properties of a node-weighted Laplacian operator and combine it with a fine-tuning mechanism related to the weighted vertex cover problem. The proposed method is applicable to large-scale networks with millions of nodes. It outperforms current state-of-the-art methods and opens more directions for understanding the vulnerability and robustness of complex systems.

The important nodes in complex networks are the extraordinary nodes which play more significant role than other nodes on the structure and function of the networks. In recent years, the reaserch on indentifying inflential nodes in complex networks has attracted much attention, because of its great theoretical significance as well as the wide range of applications. Aiming at different types of networks and motivated by different problems and applications, researchers have proposed groups of methods. This article systematically reviews more than 30 representative methods which are classified into four categories, and detailedly compares them from the aspects of computing ideas and application scenarios, and futher analyzes the strongness and weakness of each method. On this basis, this article summarizes the existing problems and outlines eight open issues as main challenges in the near future.

Empirical networks exhibit significant heterogeneity in node connections, resulting in a few vertices playing critical roles in various scenarios, including decision-making, viral marketing, and population immunization. Thus, identifying key vertices is a fundamental research problem in Network Science. In this paper, we introduce vertex entanglement (VE), an entanglement-based metric capable of quantifying the perturbations caused by individual vertices on spectral entropy, residing at the intersection of quantum information and network science. Our analytical analysis reveals that VE is closely related to network robustness and information transmission ability. As an application, VE offers an approach to the challenging problem of optimal network dismantling, and empirical experiments demonstrate its superiority over state-of-the-art algorithms. Furthermore, VE also contributes to the diagnosis of autism spectrum disorder (ASD), with significant distinctions in hub disruption indices based on VE between ASD and typical controls, promising a diagnostic role for VE in ASD assessment.

The robustness of complex networks under targeted attacks is deeply connected to the resilience of complex systems, which is defined as the ability to make appropriate response to the attack. In this paper, we study robustness of complex networks under a realistic assumption that the cost of removing a node is not constant but rather proportional to the degree of a node or equivalently to the number of removed links a removal action produces. We have investigated the state‐of‐the‐art targeted node removing algorithms and demonstrate that they become very inefficient when the cost of the attack is taken into consideration. For the case when it is possible to attack or remove links, we propose a simple and efficient edge removal strategy named Hierarchical Power Iterative Normalized cut (HPI‐Ncut). The results on real and artificial networks show that the HPI‐Ncut algorithm outperforms all the node removal and link removal attack algorithms when the same definition of cost is taken into consideration. In addition, we show that, on sparse networks, the complexity of this hierarchical power iteration edge removal algorithm is only O(n log^2+ε⁡n).