# Robustness of Cyber-Physical Supply Networks in Cascading Failures
## 6 Discussion and Conclusions
### Conclusions  
We studied the robustness of the cyber-physical supply networks against cascading failures. Our model consists of a physical supply network where the failure of a node results in flow redistribution and possible further failures due to underloading, and a cyber supply network where the failure of a node leads to flow redistribution and possible further failures due to overloading. The relationship between the cyber supply network and physical supply is one-to-one interdependence. Besides, we employed China's electric vehicle supply network and BA network to model cascading failures for cyber-physical supply networks. We obtained some meaningful results that can provide theoretical guidance to build cyber-physical supply networks with higher robustness:  
Firstly, the negative correlation between the robustness of the cyber-physical supply networks and the upper-bound capacity parameter \(\varphi\) was proved in cyber node failure. In general, cyber devices do not operate at full capacity, and a larger value of \(\varphi\) means more redundant capacity of devices and can mitigate the cascading propagation.  
Secondly, a positive relationship was observed between the lower-bound capacity and cascading size, and the size of cascading failures is mainly determined by \(\gamma\) under physical node failure. The lower-bound capacity parameter \(\gamma\) is related to the firm's operating agility and resilience. A more competitive firm is often related to a lower value of \(\gamma\).  
Thirdly, a U-shaped relationship was observed between the node load and robustness of cyber-physical supply networks under cyber or physical node failure. As the load rises, the strengthened capacity may gradually lose its effectiveness in mitigating cascading failures, that is, as the load increases, it becomes increasingly difficult to reduce cascading failures by strengthening capacity.  
In the future, we can develop cascade defense strategies to reduce the consequences of cascading failures in cyber-physical supply networks. Moreover, we will find the optimal cyber-physical supply networks with comprehensive tolerance against random and targeted failure.  
Conceptualization, D.M. and X.Y.; methodology, X.Y. and H.R.; software, X.Y.; validation, X.Y. and H.R.; formal Analysis, X.Y. and H.R.; investigation, X.Y.; resources, X.Y. and H.R.; data curation, D.M. and X.Y.; writing--original draft preparation, D.M. and X.Y.; writing--review and editing, X.Y. and H.R.; visualization, X.Y. and H.R.; supervision, D.M.; project administration, D.M.; funding acquisition, D.M. All authors have read and agreed to the published version of the manuscript.  
This work is supported by the Fundamental Research Funds for the Central Universities (No.2019 YJS072) and the National Natural Science Foundation of China (No. 71473013).  
The data used to support the findings of this study are available from the corresponding author upon request.  
The authors are thankful to the editors and the reviewers for their comments and suggestions to improve the quality of the manuscript.  
We declare that there are no conflict of interest.