# On the Robustness of No-Feedback Interdependent Networks
## Simulation and Analysis
### Coupling Strength  
This subsection is introduced to investigate the effect of \(CS\) on the robustness of no-feedback interdependent networks. We started the simulation from a special case where different subnetworks had equal coupling strength, i.e., \(CS_{A}=CS_{B}=CS\). Based on the node number and average degree set given above, the cascading failure of interdependent networks was simulated as follows. At the beginning, according to different failure modes, we randomly select nodes with a ratio of \(p\) as failure nodes in only one subnetwork or the whole interdependent networks, then the initial failures cause the cascading failure. When the cascading failure ends, the results of \(S_{A}\), \(S_{B}\), and \(S_{AB}\) with different coupling strength under two failure modes are demonstrated in Figures 2 and 3, respectively. It should be noted that \(S_{AB}=(S_{A}+S_{B})/2\) is shown since different subnetworks have the same number of nodes. Figure 3: Simulation results of cascading failure of no-feedback interdependent networks with different _coupling strength_ (\(CS\)) under bilateral failure (BF), results of one simulation. (**a**)cascading failure of subnetwork A with different \(CS\) under BF; (**b**) cascading failure of subnetwork B with different \(CS\) under BF; (**c**) cascading failure of the entire interdependent networks with different \(CS\) under BF. Figure 2: Simulation results of cascading failure of no-feedback interdependent networks with different _coupling strength_ (\(CS\)) under unilateral failure (UF), results of one simulation. (**a**)cascading failure of subnetwork A with different \(CS\) under UF; (**b**) cascading failure of subnetwork B with different \(CS\) under UF; (**c**) cascading failure of the entire interdependent networks with different \(CS\) under UF. According to the results of Figures 2 and 3, it can be concluded that the robustness of no-feedback interdependent networks under two failure modes shows a negative correlation with coupling strength. The larger the coupling strength, the worse the robustness of the interdependent networks. The reason is that if \(CS\) becomes larger, the number of nodes with dependency links is larger, there will be more failures propagate between subnetworks through dependency links, thus subnetworks are more likely to break into more components, leading to an intense cascading failure process. Moreover, when \(CS\) increases to a certain value (such as 0.8 under unilateral failure and 0.6 under bilateral failure, as shown in Figures 2 and 3), a sudden collapse (first order percolation) will occur along with the cascading failure of interdependent networks. Comparing the results of Figures 2 and 3, it was observable that the robustness under bilateral failure was worse than unilateral failure. This phenomenon is not difficult to explain. According to the definition of failure mode given in Section 2.2, if two fractions of failure nodes under unilateral and bilateral failure are equal, the number of failure nodes under bilateral failure is two times that under unilateral failure, resulting in a larger failure range and a smaller collapse threshold. A more instinctive description of the negative correlation between collapse threshold and coupling strength under unilateral and bilateral failure is given in Figure 4, where the average value of 20 independent simulations are illustrated. For each simulation, we firstly built no-feedback interdependent networks models with different \(CS\) (simulation step size of \(CS\) is 0.1), then simulated the cascading failure of those models under two failure modes (the simulation step size of failure nodes fraction was set to 0.01), and lastly, found the minimum scale of failure nodes for the complete collapse of model under different CS, that is, the collapse threshold \(f_{c}\).