# On the Robustness of No-Feedback Interdependent Networks
## Simulation and Analysis
### Dependency Strength  
This section investigates the influence of dependency strength on the robustness of no-feedback interdependent networks. Based on the parameters determined in Section 2.1, a weakly coupled no-feedback ER-ER interdependent networks model was developed in this section, where the coupling strength of both subnetworks were valued as 1 to eliminate the influence of this factor, and the dependency strength was assigned with different \(DS\). The main purpose of this experiment is to study the relationship of the size of giant component and the initial failure nodes ratio \(p\), then analyze the effect of dependency strength on the robustness. The simulation step sizes of \(DS\) and \(p\) are set to 0.01. Figure 7 illustrates the relationship between the size of the giant component and dependency strength. Specifically, Figure 7a-c demonstrate the giant component of subnetwork A, subnetwork B, and the entire interdependent networks under unilateral failure, respectively. Figure 7d-f show the results under bilateral failure. It can be seen in Figure 7 that as the dependency strength becomes larger, the cascading failure of interdependent networks experiences a moderate change from first-order to second-order phase transition under unilateral and bilateral failure, which means the robustness of the interdependent system is diminishing. Comparing the results under unilateral failure in Figure 7a,b, it was noticeable that failure in subnetwork A did not cause large scale levels of failure in B when \(DS\) was small. Specifically, if \(DS=0\), the initial failure could not propagate through dependency links between subnetworks A and B, and the failure of nodes in subnetwork A had no effect on subnetwork B. After \(DS\) increased to more than 0.7, the cascading failure of subnetworks A and B had an obvious first-order transition. In addition, the results under bilateral failure in Figure 7d,e indicate that the cascading failures of subnetworks A and B were basically the same. The explanation of this situation is that failure nodes are distributed evenly in two subnetworks under bilateral failure, whereas only one subnetwork has failure nodes under unilateral failure mode. When \(DS\) increased to 0.5, the first-order percolation phenomenon appeared, which means that the influence of bilateral failure on the robustness was greater than unilateral failure. The relationship between dependency strength and the robustness of interdependent networks is further discussed as follows. According to the definition of collapse threshold given in Section 2.2,  
Figure 7: The cascading failure of subnetwork A, subnetwork B, and the entire interdependent networks with different _dependency strength_ (\(DS\)) under unilateral failure (UF) and bilateral failure (BF), results of one simulation. (**a**) cascading failure of subnetwork A with different \(DS\) under UF; (**b**) cascading failure of subnetwork A with different \(DS\) under BF; (**c**) cascading failure of subnetwork B with different \(DS\) under UF; (**d**) cascading failure of subnetwork B with different \(DS\) under BF; (**e**) cascading failure of the entire interdependent networks with different \(DS\) under UF; (**f**) cascading failure of the entire interdependent networks with different \(DS\) under BF;  
the intersection curves between surface and horizontal plane in Figure 7a,f represent the variation of the collapse threshold with dependency strength in one simulation. Accordingly, Figure 8 summarizes the results of multiple simulations. In the processes of this simulation, we firstly construct weakly interdependent networks models different \(DS\) (simulation step size of \(DS\) is 0.1), then the latter processes are the same as that of the processes for plotting Figure 4. It is observable in Figure 8 that the collapse threshold decreased as dependency strength increased under both two failure modes.