# Vulnerability Analysis of a Multilayer Logistics Network against Cascading Failure
## 4 Vulnerability Analysis
### Structural Heterogeneity Vulnerability  
The node with the maximum load is the most critical node in the network, and it plays a vital role in the timeliness and connectivity of logistics supply and demand between cities. After these critical nodes are attacked, isolated nodes will be generated on a large scale, which will lead to a change in network topology and a fast decrease in network operation efficiency. To study whether the characteristics of cascade failure are affected by network heterogeneity, the network efficiency of different network layers is normalized and compared. As seen in the Figure 7, when only a single network-layer node is removed, the decline rate of different network efficiency curves and the failure scale of nodes are different, which means that under the same attack strategy, the disturbance effect caused by different network structures is different. This is embodied in the following three aspects: (1) Compared with the result of a single network layer, the network efficiency in a multilayer network is larger. In each network layer, node failures are caused not only by traffic redistribution but also by the load transferred by congested nodes in another network layer that has dependencies. Therefore, the iteration of multiple networks amplifies the fault of a multilayer system and easily triggers the further expansion of the failure scale. However, due to the stability of the organizational network of logistics enterprises to attack, the failure speed of the multilayer network is slowed down to some extent, so the vulnerability of the multilayer network is reduced, and high network efficiency is maintained. (2) In the organizational network of logistics enterprises, regardless of the minimum load attack or the maximum load attack, the impact on the network efficiency is relatively gentle, indicating that the attack on nodes does not easily trigger a large-scale cascading failure phenomenon. The reason is that the organizational network of logistics enterprises has a small average clustering coefficient, the network structure is relatively sparse, and the nodes are not evenly distributed. In a sparse and uneven organizational network, only a few communities have cascading failures. The attacked nodes are distributed in different communities at the same time, and the load of the failure node is difficult to transfer between different communities. Therefore, the network characteristics of small clusters enable the organizational network of logistics enterprises to maintain the efficient operation of the global network, even if a small number of independent groups are damaged. (3) The small capacity parameter causes large-scale destruction of the network, and the destruction degree is affected by the density of the network and the structure of the network. When the capacity parameter  
Figure 7: The network efficiency of different network layers varies with capacity parameters under cascade failure. \(\alpha=0.1\) is used, the damage to the international trade network is serious, and the loss of network operation efficiency is higher than 90%. Maritime networks and air transport networks came in second, with a network efficiency and failure node size of nearly 50%, indicating that the smaller node carrying capacity allows an attack to destroy almost half of the network. The operational efficiency of the organizational network and multilayer network of logistics enterprises is relatively high. According to the above analysis of network topological characteristics, the international trade network, maritime transport network and air transport network all have high small-world characteristics, while the remaining networks are relatively sparse and non-uniform. Combined with the small-world index (the ratio of the network clustering coefficient and average path length), the order of different network layers from large to small is as follows (Figure 7): \(G_{d}>G_{a}>G_{b}>G>G_{c}\). Under the maximum load attack, the network efficiency varies from large to small: \(G_{c}>G>G_{b}>G_{a}>G_{d}\).