# On the Robustness of No-Feedback Interdependent Networks
## 4 Conclusions  
In this paper, the robustness of interdependent networks with no-feedback dependency was investigated. Unlike previous studies that focused on the percolation process of no-feedback interdependent networks with different proportions of failure nodes, this paper focuses on the influence of dependency links on network robustness. Specifically, we focuses on the collapse thresholds of no-feedback interdependent networks from two characteristics of dependency links, coupling strength and dependency strength. The influences of these two parameters on the robustness were analyzed under two failure modes: unilateral failure and bilateral failure. Particularly, when studying the influence of coupling strength, we discussed two cases of the coupling strength of subnetworks in the same and different situations. The simulation results indicated that the collapse threshold of interdependent networks had an inverse relationship with coupling strength and dependency strength, especially under bilateral failure. This means that strong coupling strength or dependency strength makes no-feedback interdependent networks less robust. For unilateral failure, if coupling strength or dependency strength increased to more than 0.5, the robustness of the model decreased sharply, while the collapse threshold rapidly reduced from 0.7 to about 0.4. In addition, we also found that when considering only one factor, the influence of coupling strength and dependency strength on robustness was equivalent. All these results provide references for designing a robust interdependent system. If we want to construct a robust no-feedback interdependent system, it is a feasible way to reduce coupling strength or dependency strength through various protective measures. Although several achievements have been obtained, there are still some shortcomings in this research. For example, we studied the direction of the dependency link between subnetworks but did not consider the direction of the connection link in the subnetwork. While considering the direction of dependency link and connection link at the same time, the robustness of interdependent networks can be considered to be a notable problem. In addition, during the analysis of the effect of dependency strength, we did not consider the difference between different dependency links and set a fixed \(DS\) to those links. Finally, we chose to study the coupling strength parameter and dependency strength, respectively. In fact, there are many real-world interdependent systems with partial and weak dependency at the same time. To gain a better understanding of the robustness of the no-feedback interdependent system, further research on these topics will be extremely essential. The following are available online at [[http://www.mdpi.com/2076-3417/8/5/835/s1](http://www.mdpi.com/2076-3417/8/5/835/s1)]([http://www.mdpi.com/2076-3417/8/5/835/s1](http://www.mdpi.com/2076-3417/8/5/835/s1)), Figure S1: Relationship between collapse threshold fc and coupling strength (CS) under both unilateral and bilateral failure modes, each point is obtained by averaging over simulation on 20 independent realizations. Figure S2: The three-dimension curved surface shows _collapse threshold fc_ as a function of _coupling strength(CS)_ under unilateral failure, each point is averaged over 20 different network realizations. Figure S3: The three-dimension curved surface of _collapse threshold fc_ as a function of _coupling strength_ under bilateral failure, each point is averaged over 20 different network realizations. Figure S4: Collapse threshold fc as a function of _dependency strength(DS)_, each point is obtained by averaging over simulation on 20 independent realizations. [PERSON] and [PERSON] conceived and designed the experiments; [PERSON] performed the experiments; [PERSON] and [PERSON] helped in the experiment; [PERSON] analyzed the data; [PERSON] and [PERSON] wrote the paper. This work was supported by the National Natural Science Foundation of China under Grant No. 61603408.