Mining key nodes in complex networks has always been a promising research direction in the field of complex networks. Many precise methods proposed by researchers for mining influential special nodes in networks have been widely applied in a plethora of fields. However, some important node-mining methods often use the degree as a node attribute indicator for evaluating node importance, while the clustering coefficient, as an important attribute of nodes, is rarely utilized. Some methods only consider the global position of nodes in the network while ignoring the local structural information of nodes in special positions and the network. Hence, this paper introduces a novel node centrality method, KCH. The KCH method leverages K-shell to identify the global position of nodes and assists in evaluating the importance of nodes by combining information such as structural holes and local clustering coefficients of first-order neighborhoods. This integrated approach yields an enhanced performance compared to existing methods. We conducted experiments on connectivity, monotonicity, and zero models on 10 networks to evaluate the performance of KCH. The experiments revealed that when compared to the collective influence baseline methods, such as social capital and hierarchical K-shell, the KCH method exhibited superior capabilities in terms of collective influence.

In research on complex networks, mining relatively important nodes is a challenging and practical work. However, little research has been done on mining relatively important nodes in complex networks, and the existing relatively important node mining algorithms cannot take into account the indicators of both precision and applicability. Aiming at the scarcity of relatively important node mining algorithms and the limitations of existing algorithms, this paper proposes a relatively important node mining method based on distance distribution and multi-index fusion (DDMF). First, the distance distribution of each node is generated according to the shortest path between nodes in the network; then, the cosine similarity, Euclidean distance and relative entropy are fused, and the entropy weight method is used to calculate the weights of different indexes; Finally, by calculating the relative importance score of nodes in the network, the relatively important nodes are mined. Through verification and analysis on real network datasets in different fields, the results show that the DDMF method outperforms other relatively important node mining algorithms in precision, recall, and AUC value.

Identifying critical edges in complex networks is a fundamental challenge in the study of complex networks. Traditional approaches tend to rely solely on either global information or local information. However, this dependence on a single information source fails to capture the multi-layered complexity of critical edges, often resulting in incomplete or inaccurate identification. Therefore, it is essential to develop a method that integrates multiple sources of information to enhance critical edge identification and provide a deeper understanding and optimization of the structure and function of complex networks. In this paper, we introduce a Global–Local Hybrid Centrality method which integrates a second-order neighborhood index, a first-order neighborhood index, and an edge betweenness index, thus combining both local and global perspectives. We further employ the edge percolation process to evaluate the significance of edges in maintaining network connectivity. Experimental results on various real-world complex network datasets demonstrate that the proposed method significantly improves the accuracy of critical edge identification, providing theoretical and methodological support for the analysis and optimization of complex networks.

Reward is an effective mechanism that promotes cooperation. However, an individual usually reward her opponents in certain cases. Inspired by this, a conditional neutral reward mechanism has been introduced. In detail, an individual will reward his or her neighbors with the same strategy when the payoff of the focal one is higher than that of his or her neighbors. And simulations are conducted to investigate the impact of our mechanism on the evolution of cooperation. Interestingly, cooperation can survive and dominate the system. Nominal antisocial reward that defectors reward each other is rarely because of the greed of defectors. By contrast, cooperators inside the cooperative clusters share the payoff with cooperators on the boundary so that the latter can form shields to protect cooperators.

The heterogeneous structure implies that a few nodes may be crucial in maintaining network structural and functional properties. Identifying these crucial nodes correctly and quickly is a primary issue as contemporarily may face the mushrooming of large-scale datasets. Besides, the ‘weight issue’ is always ignored in this field which edge weight may play a positive/negative role in contributing to the node importance in different weighted networks. This paper provides a novel algorithm, Weighted Expectation Algorithm (WEA), which aims to ensure accuracy and speed of computation by taking advantage of dynamic programming to better handle the task of large-scale networks. Additionally, the weight issue that edge weights may contribute differently is addressed by a simply quantitative definition. Two standard experiments show WEA can maintain the network structure in connectivity well (by the lowest average robustness 0.192) and identify the node importance better in the spreading function test of spreading dynamics (by the highest average Kendall’s tau-b 0.678). In addition, the time complexities of different algo-rithms are evaluated, and their time-consuming are tested, proving that WEA consumes a rela-tively short time.