## 2 Related Works  
WSNs are usually deployed in unattended environments and faced with varied potential attacks, resulting in the increased probability of node failure [16; 17]. Thus, the invulnerability improvement has become a vital issue for the practicality and feasibility of WSNs. Due to the restriction of cost and technique, it is impossible to enhance network invulnerability by entirely depending on optimizing node capability. At the same time, improving invulnerability through node backup will seriously boost network redundancy and energy consumption [18]. Therefore, promoting the stability and reliability of WSNs by using efficient network routing schemes without raising the fixed investment has become a trending topic. The measurement of network invulnerability is undoubtedly a quantitative index to evaluate QoS and is the key to network invulnerability inquisition [19]. Recently, the most widely used invulnerability indicators of non-topological WSNs primarily include network lifetime, survival ratio, and network coverage, etc. [20; 21; 22]. These indicators are easy to obtain, so one or more indicators are usually used to estimate invulnerability. Since these indexes mainly employ network attributes unrelated to topology structure as the evaluation criteria, these network attributes cannot actually and accurately depict invulnerability [23]. On this basis, the topology-based invulnerability indexes come into being. These indexes apply network connectivity as the object, adjacency matrix as the source of feature extraction, graph theory and probability statistics as theoretical means to compute invulnerability more comprehensively [24; 25]. For instance, [PERSON] introduced the graph toughness to assess the invulnerability performance of WSNs [26]. Although there are numerous quantitative studies on topology-based invulnerability, the methods focus on various description perspectives and have specific limitations for different types of networks under the assumption of stationary network topology. However, in the application of hierarchical routing protocols, the clustering structure varies dynamically, resulting in the failure to estimate the invulnerability of whole network in variable environments [27]. The evaluation method based on the equivalent shortest path number can describe the dynamic communication process through the shortest path after the network impairment and can be used to characterize the local invulnerability of network [28]. Moreover, on the basis of connectivity, considering the characteristics of network convergence, the invulnerability measure method using effective connectivity is biased towards the invulnerability of overall network [29]. Thus, it is definitely valuable to combine the invulnerability indicator based on the shortest path number and the invulnerability metric based on the connectivity to clarify the general survivability of network. Scale-free topology has been used by multiple researchers to improve the invulnerability of WSNs under external attacks [30; 31; 32; 33]. However, these studies fail to precisely assess the impact of different node failure scenarios and damage types on invulnerability performance. In addition, these studies mainly concentrate on appraising network invulnerability, neglecting the analysis and optimization of network structure. Moreover, the analysis of invulnerability is undoubtedly concerned with the guarantee of QoS performance under the impact of various network attacks [34; 35; 36; 37]. Nevertheless, the related studies cannot fully reflect the difference value of network invulnerability in the process of node failure. Furthermore, it can effectively indicate the dynamic variation of invulnerability by describing the change in network performance caused by removing critical nodes [38]. However, the existing attack schemes rarely involve the evaluation of node importance, which is usually called centrality.