Abstract: A complex network is a logical model that can reflect the connectivity between substances. It abstracts the entities in a complex system into nodes, and the relationship between entities into edges. Complex network systems are closely related to people's lives and have been widely used in transportation systems, communication systems, power systems, industrial systems and other fields. If the nodes and edges in the network are affected by external environments such as natural disasters and human attacks, and failures occur, this will bring multiple losses to production and life, such as human, material and financial resources. Therefore, in order to reduce the losses, it is of great significance to study the prioritized recovery order of failed nodes in complex network systems to improve the performance recovery ability of complex systems network after failure.
Aiming at the problem of node recovery order after multi-node failure in complex network systems, in order to study the recovery priority of failed nodes from different perspectives, this paper establishes the loss importance measure, recovery importance measure and resilience importance measure models of complex network systems based on Birnbaum importance measure theory, which combines the loss and recovery of complex network performance with the importance. These models consider the impact of node state changes on both loss and recovery of system performance, so that the critical nodes with the greatest impact on system performance can be found, and the recovery priority of the failed nodes can be studied.In this paper, firstly the performance change process of the complex network system after being hit and repaired is analyzed, and the loss performance and recovery performance of the complex network system are defined. Then the network nodes are divided into two states, normal operation and failure, and the loss performance and recovery performance of the system are analyzed respectively when the nodes are in different states. Different nodes also have different impacts on network performance, and the loss importance measure of nodes and the recovery importance measure of nodes are defined by combining the performance with the importance measure theory, respectively. By comparing the recovery importance value of each node, the recovery of network performance can be evaluated. By comparing the loss importance value of each node, the change of network vulnerability can be assessed, so that preventive maintenance can be carried out in advance for nodes with higher importance. Finally, the ratio of recovery importance measure to loss importance measure of a node is defined as the resilience importance measure of the node.Resilience importance measure integrates the impact of the node state on both the loss and recovery of system performance, and the larger the value of the node's resilience importance measure, indicating that the node has a greater impact on the network performance, the higher the node's maintenance priority.
In order to verify the validity of the proposed importance measure model, this paper introduces a land transportation network system containing 6 transportation aggregation points and 11 transportation dispersion points. The nodes in the land transportation network are divided into transportation hubs and transportation dispersal points, and the connecting edges indicate the transportation routes between the transportation nodes. The land transportation network flow is represented by the cargo volume of nodes and connecting edges, and it is assumed that the states of all transportation nodes and routes in the land transportation network are independent of each other. By substituting the cargo volume of different transportation nodes in different states into the importance measure model, the loss importance measure value, recovery importance measure value and resilience measure importance value of the nodes are obtained, respectively, so that the impact of the node state on the vulnerability of the system, the system recovery situation and the system resilience can be assessed, and the recovery sequence of the failed nodes in different states can be obtained. Ultimately, by comparing the importance measure values, the recovery order of different failed nodes can be obtained. The case of the land transportation network illustrates, to some extent, the validity of the importance measure models proposed in this paper. However, these importance measure models still have shortcomings, and the model consideration conditions are not comprehensive enough. Since the recovery cost of different nodes may also be different, and the nodes that have a large impact on the recovery of system performance may also have a higher recovery cost. In future research we can consider the cost into the model and study the impact of failed nodes of network systems on the performance recovery of complex networks under the inclusion of recovery cost constraints. In addition, this paper only studies the recovery order of failed nodes, and in the future, we can also study the importance measure and recovery of failed edges.

Key words: reliability, network system, importance measure, node failure

摘要： 复杂网络系统中的节点易受自然灾害和人为攻击等外部环境的影响而失效,这会给人们的生产生活带来不必要的损失。为了降低损失,研究复杂网络系统中失效节点的优先恢复顺序具有重要意义。本文提出了衡量网络节点重要性的损失重要度、恢复重要度和韧性重要度模型。首先,分析了复杂网络遭受外部攻击后的系统性能下降情况和维修失效节点后的系统性能恢复情况。然后,基于节点运行状态对系统性能变化的影响,提出了节点的损失重要度和恢复重要度,并且基于节点的损失和恢复情况提出了衡量失效节点恢复性的节点韧性重要度。最后,以包含6个运输集散点和 11 个运输分散点的陆运网络为例,分析了不同节点失效和恢复时的陆运网络系统性能变化情况,通过比较重要度值的大小,可以得到不同失效节点的恢复顺序,验证所提模型的有效性。

关键词: 可靠性, 网络系统, 重要度, 节点失效