# On the Robustness of No-Feedback Interdependent Networks
## Simulation and Analysis
### Coupling Strength
Calculating the average value of 20 thresholds under the same CS, we can plot the curves in Figure 4. For unilateral failure, when \(CS\) as less than 0.4, the collapse threshold of interdependent networks was maintained at a level of about 0.7. The reason is that when \(CS\) is small, failure propagation between the subnetworks through the dependency link is very limited after initial failure, therefore the threshold does not change much. However, in the range of 0.4 to 1, the collapse threshold decreased sharply as \(CS\) increased. At the same time, the cascading failure scale caused by the initial failure kept expending along with the increase of \(CS\). When the parameter \(CS\) reached 1, the no-feedback interdependent networks model was transformed into fully interdependent networks, and the collapse threshold was reduced to 0.39, which is consistent with Reference [1]. In addition, the collapse threshold under bilateral failure decreased faster than unilateral failure as coupling strength increased, and finally ended at about 0.23 under full dependency. Therefore, it is safe to conclude that no-feedback interdependent networks under bilateral failure are more sensitive to the change of coupling strength. Figure 4: Relationship between collapse threshold \(f_{c}\) and _coupling strength_ (\(CS\)) on both unilateral and bilateral failure modes. Based on the experiments for \(CS_{A}=CS_{B}\) given above, a more general scenes \(CS_{A}\
eq CS_{B}\) will be investigated in the following. The simulation process is similar to the process for plotting Figure 4, and the only difference is setting different coupling strength parameters \(CS_{A}\) and \(CS_{B}\) to subnetworks A and B. Therefore, the results of this simulation were plotted on a three-dimensional curved surface. The result of the collapse threshold of subnetwork A under unilateral failure as a function of coupling strength is shown in Figure 5. On the other hand, the result of collapse threshold of the whole interdependent networks under bilateral failure is shown in Figure 6. It is worthwhile pointing out that curves in Figure 4 are special cases (\(CS_{A}=CS_{B}\)) in Figures 5 and 6. For exemplification, they are highlighted with gray dotted lines. In Figure 5, it is observable that the collapse threshold changed slightly under unilateral failure when coupling strength was small, but the robustness of interdependent networks declined rapidly if the coupling strength was more than 0.5. As shown in Figure 6, the collapse threshold under bilateral failure decreased more intensely than that under unilateral failure. With the coupling strength \(CS_{A}\) and \(CS_{B}\) increased from zero to one, the collapse threshold decreased gradually, and it reached 0.23 when \(CS_{A}=CS_{B}=1\). Figure 5: The three-dimensional curved surface of _collapse threshold_\(f_{c}\) over various _coupling strength (CS)_ under unilateral failure, each point is an average of 20 different network realizations. Sub-graph (**a**-**e**) show the cascading failure under specific coupling strength in one simulation. Figure 6: The three-dimensional curved surface of _collapse threshold_\(f_{c}\) over various _coupling strength_ under bilateral failure, each point is an average of 20 different network realizations. It should be noted that several special cases are marked with circles in Figure 5. They are discussed as follows:  
1. If \(CS_{A}=1\), \(CS_{B}=0\) or \(CS_{A}=0\), \(CS_{B}=1\), one subnetwork is fully dependent on the other subnetwork through directed dependency links. The cascading failure of subnetworks and interdependent networks are shown in Figure 5a,d. 2. If \(CS_{A}=CS_{B}=1\), two subnetworks are fully coupled by bidirectional dependency links, which are equivalent to the model with one-to-one correspondence in [1]. The failure process shows a second-order transition phenomenon in Figure 5b. 3. If \(CS_{A}=CS_{B}=0\), two subnetworks have no dependency on each other, so failure in one subnetwork will not spread to the other.