## 7 Conclusions  
In this paper, the invulnerability-aware clustering routing algorithm based on game-theoretic method is proposed to improve invulnerability and balance network energy consumption in WSNs. In the network, each node in an individual cluster achieves the optimal probability of becoming a CH via the game of invulnerability and residual energy. In addition, the dominance function for candidate CHs considering the remaining energy and the distance from the BS is constructed to avoid the scenario of multiple CHs in a cluster. A cluster update mechanism is established to further optimize the network energy consumption by rotating the nodes with excessive energy consumption to new clusters. Furthermore, the network invulnerability under three external attacks rises first and then declines with varied network parameters. Specifically, the invulnerability of maximum-degree attack is the highest, indicating that the addressed routing algorithm has the most robust capability to reduce the negative impact on invulnerability performance of maximum-degree attack. With the change of network parameters, the average residual energy of random attack decreases the slowest, signifying that the presented routing algorithm can achieve the most satisfactory energy performance when encountering random attack. At the same time, increasing the number of CH elections \(r\) is useful to significantly boost the number of communication rounds.  
For the proposed routing protocol, CH election computation and intra-cluster communication overhead mainly concentrated in the clustering setup and maintenance phase. Therefore, in future research, to reduce energy consumption, more representative nodes should be preliminarily chosen to participate in the CH election according to the distance from the BS, residual energy, node centrality, etc. Meanwhile, the weight of invulnerability and residual energy in the profit function should be adaptively adjusted to obtain more acceptable network performance under network attacks. In addition, IACRA is superior to GEEC, HGTD, and EEGC in terms of extending the network life cycle, improving the network invulnerability and balancing the network energy consumption. Nevertheless, the feasibility and effectiveness of the routing protocol in other network topologies still require verification. Furthermore, in the case of introducing new adversaries, rebuilding the profit matrix, modifying the existing game model, and reconducting a Nash equilibrium are undoubtedly demanded.  
Conceptualization, J.W.; methodology, J.W. and Y.Z.; software, B.L.; validation, B.L. and P.M.; formal analysis, Y.Z.; investigation, J.W. and C.H.; resources, J.W.; data curation, P.M. and Y.Z.; writing--original draft preparation, J.W., Y.Z. and C.H.; writing--review and editing, Y.Z. and C.H.; visualization, Y.Z.; supervision, J.W.; project administration, J.W.; funding acquisition, J.W. and P.M. All authors have read and agreed to the published version of the manuscript.  
This research was funded by National Natural Science Foundation of China (Grant no. 61771184), Program for Science & Technology Innovation Talents in Universities of Henan Province (Grant no. 20 HASITIT029), Key Scientific Research Projects in Universities of Henan Province (Grant no. 19A460021), Major Special Science and Technology Project of Luoyang (Grant no. 2101018A), and Key Science and Technology Project of Henan Province (Grant no. 222102102164).  
Not applicable.  
Data Availability Statement: The data used to support the findings of this study are available from the corresponding author upon request.  
The comments and suggestions of our colleagues in Henan University of Science and Technology are highly appreciated.  
The authors declare no conflict of interest.