## 5 A Case Study in Engineering Application
### The Effectiveness of the Resilient Recovery Method based on Outer Synchronization  
Assuming that a failure occurs in the enterprise node \(W_{1}\), this causes a pause of the contacts in \(W_{1}\_M_{10}\), \(W_{1}\_M_{11}\), and \(W_{1}\_P_{2}\). In this disruption, the network structure is changed, along with the status values and the parameters. The supply network \(B\) acts as the response network, where \(B\
eq A\). Assuming that \(\overline{\alpha}(m,r,b)\) are unknown after the disruption, the status of the response network is initialized randomly. An adaptive impulsive controller is adopted in Section 3. Assuming that the period of impulse control \(t=0.05\), the feedback matrix \(B_{\bar{k}}=diag[-0.5,\,-0.5,\,-0.5]\), the inner coupling matrix \(\Gamma=diag(0.2,\,0.2,\,0.2)\), and the controller's parameters are \(L=1\), \(g=7\), \(\mu=1.5\), \(t_{k}=0.05\). By using MATLAB Toolbox, we can get \(\kappa=2.8419\), \(\iota=3.0448\), \(\lambda_{\max}=0.25<1\), \(v=(L-g+\kappa+\iota)=-0.1133<0\), and \(\chi=\max\{\
u,-\mu)=-1.5<0\). \(\exists\xi>1\), such that \(2\chi\tau_{k}+\ln(\xi)<0\). Then, Theorem 1 can be satisfied and the network can achieve outer synchronization. The synchronization errors are shown in Figure 9. Here, \(e_{i}\to 0\) after 3 impulse periods; \(e_{i}(t)\) reduces obviously at times \(t=0.05\), \(t=0.1\), and \(t=0.15\). This means that the impulsive control can recover rapidly. The parameter identification of the SN is realized in Figure 10. The uncertain parameter \(\overline{\alpha}\) in the actual network (as in Equation (4)) will adjust adaptively in the process of outer synchronization and approach \(\alpha(m=10;\,r=2.6;\,b=28)\) in the scheduled network (Equation (3)). The parameter  
Figure 8: Structure of a European supply network \(A\). Figure 9: Error \(e_{i}(t)\). We start again from the initial scheduled network \(A\) and remove these links in random order. This shows that the removal sequence does not significantly influence the resilient recovery ratio \(R(t)\), confirming that these links are redundant for the resilient recovery. Furthermore, experiments show that the backbone of the network recovery is related to the initial value. That is to say, the different initial values will obtain different network backbones. In order to show the effect of the backbone identification on the resilient recovery, the following cases are compared, as shown in Table 4. In the experiments, the above initial value is adopted, yet the resilient recovery ratio \(R(0.3)\) at time \(t=0.3\) in different disrupted networks is calculated. Case 1: In network 1, a disruption occurs in the enterprise nodes\(W_{1}\) and \(W_{3}\), causing six links belong to the backbone to be removed. In network 2, a disruption still occurs in the enterprise nodes\(W_{1}\) and \(W_{3}\), causing six redundant links (not belonging to the backbone) to be removed, \(R\) (network 1) \(<R\) (network 2), as shown in Figure 14.  
Case 2: In network 3, a disruption occurs in the enterprise nodes\(W_{2}\) and \(P_{2}\), causing six links that belong to the backbone to be removed. In network 4, a disruption still occurs in the enterprise nodes\(W_{2}\) and \(P_{2}\), causing six redundant links to be removed, \(R\) (network 3) \(<R\) (network 4), but the difference is not obvious. Case 3: In network 5, a disruption occurs in the enterprise nodes\(W_{1}\) and \(W_{5}\), causing six links that belong to the backbone to be removed. In network 6, a disruption still occurs in the enterprise nodes\(W_{1}\), \(W_{5}\), and \(P_{3}\), causing six redundant links to be removed, \(R\) (network 5) \(<R\) (network 6), where the difference is obvious. In networks 2, 4, and 6, the removal of six redundant links has little effect on resilient recovery, despite their different locations. In networks 1, 3, and 5, the removal of 6 backbone links exerts greatly effect on resilient recovery.