Many networks in the real-world have spatial attributes, such as the location of nodes and the length of edges, called spatial networks. When these networks are subject to some random or deliberate attacks, some nodes in the network fail, which causes a decline in the network performance. In order to make the network run normally, some of the failed nodes must be recovered. In the case of limited recovery resources, an effective key node identification method can find the key recovering node in the failed nodes, by recovering which the network performance can be recovered the most. In this paper, we propose two key recovering node identification methods for spatial networks, which are the Euclidean-distance (ED) recovery method and route-length (RL) recovery method, respectively. Simulations on homogeneous and heterogeneous spatial networks show that the proposed methods can significantly recover the network performance.

As important infrastructure, logistic networks need to be designed not only for robustness but also for transportation efficiency. In order to improve transportation efficiency, different types of logistic networks integrate to form a double-layer coupled network. When some nodes fail in this double-layer coupled network, especially in the case of limited repair resources, how to evaluate the node that needs to give priority to repair is of great significance. In this study, an evaluation method of key repairing node is proposed to find the key node which should be repaired first to restore the network performance. By comparing with traditional evaluation methods of key nodes, the effectiveness of the proposed method is verified.

Traditional methods to identify the important nodes are suitable for single networks. However, many real-world networks are coupled together, which can be modeled by multi-layer networks. Therefore, traditional identification methods may not be suitable for multi-layer networks. In this paper, we propose a new method to identify the important nodes in multi-layer logistic network. Considering the dynamic of the network, a new routing strategy based on the greedy algorithm and iterative method is proposed. The traditional betweenness centrality and closeness centrality are modified according to the new routing strategy to show the traffic condition and topology characteristics of each node. Then the new identification method is proposed based on the modified betweenness and closeness. The new method is compared with some traditional ones, and the simulation results show its advantages.