Vital nodes identification in complex networks

Vital nodes identification, that is, finding a set of nodes whose absence would cause a collapse of the network, is a significant project in network science. Despite there are a plenty of methods to identify the vital nodes, two major problems still need to be solved, that is how to select these nodes and how to determine the number of them. In this study, we focus on dealing with these two problems via proposing a multiobjective memetic algorithm for vital nodes identification task. First, vital nodes identification task is modeled as a biobjective optimization problem by analyzing the characteristic of vital nodes. Then, a memetic strategy and specific evolutionary operators inspired by multiple centralities are designed to execute local and global search. In addition, a long-tail property is found from the Pareto front of this bi-objective optimization problem, and the simulation results always show an obvious knee region. Hence, an adaptive learning method to determine the size of vital nodes is designed by searching for the knee point. At last, the proposed framework is tested on the scale free networks and real-life networks, and the simulation results validate its effectivity in contrast with the state-of-art greedy methods.

Vital layer nodes of multiplex networks for immunization and attack

Identifying influential nodes to maximize the spread of information within networks is a vital combinatorial optimization problem with extensive practical applications. Unlike proposing a specific node ranking method to identify vital nodes, this study introduces CycRank, a universal framework to optimize the strategies for selecting vital nodes in existing methods by leveraging cycle structures. The experimental results demonstrate that, compared to directly selecting top-k nodes from centrality rankings and state-of-the-art optimization frameworks, the influencers identified by CycRank increase the average dissemination range by up to 17%. Additionally, regardless of the centrality measures or network types, these influencers exhibit lower degree and greater average distances, effectively striking a delicate trade-off between their influence, dispersion, and hub properties. Our study not only paves the way for novel strategies in vital nodes identification but also underscores the unique potential of underappreciated cycle structures.

Identifying vital nodes is one of the core issues of network science, and is crucial for epidemic prevention and control, network security maintenance, and biomedical research and development. In this paper, a new vital nodes identification method, named degree and cycle ratio (DC), is proposed by integrating degree centrality (weight α) and cycle ratio (weight 1 − α). The results show that the dynamic observations and weight α are nonlinear and non-monotonicity (i.e., there exists an optimal value α* for α), and that DC performs better than a single index in most networks. According to the value of α*, networks are classified into degree-dominant networks (α* > 0.5) and cycle-dominant networks (α* < 0.5). Specifically, in most degree-dominant networks (such as Chengdu-BUS, Chongqing-BUS and Beijing-BUS), degree is dominant in the identification of vital nodes, but the identification effect can be improved by adding cycle structure information to the nodes. In most cycle-dominant networks (such as Email, Wiki and Hamsterster), the cycle ratio is dominant in the identification of vital nodes, but the effect can be notably enhanced by additional node degree information. Finally, interestingly, in Lancichinetti–Fortunato–Radicchi (LFR) synthesis networks, the cycle-dominant network is observed.

Vital nodes play a pivotal part of network structure and dynamics, where finding the minimal size of a set of vital nodes belongs to an NP-hard problem and cannot be solved by a polynomial algorithm. Recent studies of vital nodes identification mostly rely on the structural information, such as collective influence and degree. However, the performance of local-based methods varies for different structure, while the complexities of global-based methods are generally high for most situations. Here, we map the problem into an optimization issue based on global information of network structure and propose a belief propagation and node reinsertion (BPR) method with almost linear time complexity, where finding the minimum feeding back vertex set is a key. Compared with several state-of-the-art heuristic methods, the BPR method has advantages of high accuracy and practicability of vital nodes identification and low computational complexity. Under two attack schemes: static and dynamical, extensive experiments of Erdos-Rényi and scale-free models and real-world networks of the power grid and traffic network convincingly demonstrate that the BPR method remarkably outperforms other methods in vital nodes identification. This helps to reassess the operational risk of a network and improve robustness ranging from network design schemes, protection strategies to failure mitigation.

Over the most recent twenty years, network science has bloomed and impacted different fields such as statistical physics, computer science, sociology, and so on. Studying the percolation behavior of a network system has a very important role in vital nodes identification, ranking, network resilience, and propagation behavior of networks. When a network system undergoes failures, network connectivity is broken. In this perspective, the percolation behavior of the giant connected component and finite-size connected components is explored in depth from the macroscopic and meso-microscopic views, respectively. From a macro perspective, a single network system always shows second-order phase transitions, but for a coupled network system, it shows rich percolation behaviors for various coupling strength, coupling patterns and coupling mechanisms. Although the giant component accounts for a large proportion in the real system, it cannot be neglected that when the network scale is large enough, the scale of finite-size connected components has an important influence on network connectivity. We here systematically analyze the phase transition behaviors of finite-size connected components that are different from the giant component from a meso-microscopic perspective. Studying percolation behaviors from the macro and meso-micro perspectives is helpful for a comprehensive understanding of many fields of network science, such as time-series networks, adaptive networks, and higher-order networks. The intention of this paper is to provide a frontier research progress and promising research direction of network percolation from the two perspectives, as well as the essential theory of percolation transitions on a network system.

Numerous well-known processes of complex systems such as spreading and cascading are mainly affected by a small number of critical nodes. Identifying influential nodes that lead to broad spreading in complex networks is of great theoretical and practical importance. Since the identification of vital nodes is closely related to propagation dynamics, a novel method DynamicRank that employs the probability model to measure the ranking scores of nodes is suggested. The influence of a node can be denoted by the sum of probability scores of its i order neighboring nodes. This simple yet effective method provides a new idea to understand the identification of vital nodes in propagation dynamics. Experimental studies on both Susceptible-Infected-Recovered and Susceptible-Infected-Susceptible models in real networks demonstrate that it outperforms existing methods such as Coreness, H-index, LocalRank, Betweenness, and Spreading Probability in terms of the Kendall τ coefficient. The linear time complexity enables it to be applied to real large-scale networks with tens of thousands of nodes and edges in a short time.

Fake news causes an adverse effect on the regular public order and has become easier to propagate with the popularity of online social networks. The threat of fake news propagation makes it important to explore the vital nodes, which are defined as nodes with large branch sizes and hence generating a wider influence than others in this work. Previous studies about identifying vital nodes are mainly from single propagation of fake news networks, which do not consider that users may participate in different propagation networks. Here we identify vital nodes with the feature named the Ck-value that combines structural feature out-degree in a single network and multi-network user activeness. The Ck-value could reflect the branch size with a strong correlation, even at the early stage of propagation, and percolation based on Ck-value is more efficient than other indicators such as node activeness and out-degree. Thus, this research may provide a better understanding of vital nodes in the fake news propagation from topology properties, and further inspires innovative ways to identify vital nodes of fake news propagation.

Disruptive coefficient and 2-step disruptive coefficient: Novel measures for identifying vital nodes in complex networks

Measuring centrality has recently attracted increasing attention, with algorithms ranging from those that simply calculate the number of immediate neighbors and the shortest paths to those that are complicated iterative refinement processes and objective dynamical approaches. Indeed, vital nodes identification allows us to understand the roles that different nodes play in the structure of a network. However, quantifying centrality in complex networks with various topological structures is not an easy task. In this paper, we introduce a novel definition of entropy-based centrality, which can be applicable to weighted directed networks. By design, the total power of a node is divided into two parts, including its local power and its indirect power. The local power can be obtained by integrating the structural entropy, which reveals the communication activity and popularity of each node, and the interaction frequency entropy, which indicates its accessibility. In addition, the process of influence propagation can be captured by the two-hop subnetworks, resulting in the indirect power. In order to evaluate the performance of the entropy-based centrality, we use four weighted real-world networks with various instance sizes, degree distributions, and densities. Correspondingly, these networks are adolescent health, Bible, United States (US) airports, and Hep-th, respectively. Extensive analytical results demonstrate that the entropy-based centrality outperforms degree centrality, betweenness centrality, closeness centrality, and the Eigenvector centrality.

Vital nodes identification is the problem of identifying the most significant nodes in complex networks, which is crucial in understanding the property of the networks and has applications in various fields such as pandemic controlling and energy saving. Traditional methods mainly focus on some types of centrality indices, which have restricted application cases. To improve the flexibility of the process and enable simultaneous multiple nodes mining, a deep learning-based vital nodes identification algorithm is proposed in this study, where we train the influence score of each node by using a set of nodes to approximate the rest of the network via the graph convolutional network. Experiments are conducted with generated data to justify the effectiveness of the proposed algorithm. The experimental results show that the proposed method outperforms the traditional ways in adaptability and accuracy to recover the dynamical process of networked system under different classes of network structure.

An improvement method for degree and its extending centralities in directed networks

Complex networks in the real world are often with heterogeneous degree distributions. The structure and function of nodes can vary significantly, with vital nodes playing a crucial role in information spread and other spreading phenomena. Identifying and taking action on vital nodes enables change to the network’s structure and function more efficiently. Previous work either redefines metrics used to measure the nodes’ importance or focuses on developing algorithms to efficiently find vital nodes. These approaches typically rely on global knowledge of the network and assume that the structure of the network does not change over time, both of which are difficult to achieve in the real world. In this paper, we propose a localized strategy that can find vital nodes without global knowledge of the network. Our joint nomination (JN) strategy selects a random set of nodes along with a set of nodes connected to those nodes, and together they nominate the vital node set. Experiments are conducted on 12 network datasets that include synthetic and real-world networks, and undirected and directed networks. Results show that average degree of the identified node set is about 3–8 times higher than that of the full node set, and higher-degree nodes take larger proportions in the degree distribution of the identified vital node set. Removal of vital nodes increases the average shortest path length by 20–70% over the original network, or about 8–15% longer than the other decentralized strategies. Immunization based on JN is more efficient than other strategies, consuming around 12–40% less immunization resources to raise the epidemic threshold to tau sim 0.1. Susceptible-infected-recovered simulations on networks with 30% vital nodes removed using JN delays the arrival time of infection peak significantly and reduce the total infection scale to 15%. The proposed strategy can effectively identify vital nodes using only local information and is feasible to implement in the real world to cope with time-critical scenarios such as the sudden outbreak of COVID-19.

With the widespread application of complex network theory, identifying vital nodes is an important part of complex network analysis, which has been a key issue in analyzing the characteristics of network structure and functions. Although many centrality measures have been proposed to identify vital nodes, there is still a lack of an objective performance testing platform for evaluating vital nodes identification algorithms. This study introduces the complex scheduling network as an objective performance testing platform where the identification of vital nodes directly determines the decision-making in each decision scenario through the development of heuristic algorithms upon centrality measures, and therefore, the optimization goals of scheduling problems can be regarded as an objective index for evaluating vital node identification algorithms. Finally, the undirected network obtained by the open shop scheduling problem is taken as an example to analyze and evaluate the performance of various vital node identification algorithms.

Vital nodes play a pivotal role in spreading on complex networks. Various node centrality methods can quantify each node's importance to find important nodes. This paper proposed a comprehensive node influence method, which considers both node self-influence factors and total influence received from other nodes. Six real networks are employed to validate the proposed method's effectiveness. Node spreading ability in Susceptible-Infected-Recovered simulation is used as a benchmark of node importance by comparing Kendall's correlation coefficient between node centrality value and spreading ability in Susceptible-Infected-Recovered simulation to evaluate the performance of different centrality methods on finding essential nodes. In the simulation, the proposed method has the highest Kendall's correlation coefficient in all networks, and comprehensive node influence is closer to revealing the relative size of node spreading ability than existing centrality methods. The proposed method's general applicability and ability to find important nodes under different infection probabilities in the Susceptible-Infected- Recovered model are also discussed by comparing it with other centrality methods. The result shows that Kendall's correlation between comprehensive node influence and spreading ability is still advantageous over other centrality methods. Formed by node degree, K-Shell value, and topological distance between nodes, comprehensive node influence is a simple and effective way to identify vital nodes in complex networks.