# Dynamical Recovery of Complex Networks under a Localised Attack
## 2 Model and Results
Figure 3d compares the dynamic behaviours under RA and LA, and the critical value of an LA is always larger than the one under an RA. Under the same conditions, whether it is a static or a dynamic recovery for the RR network, an LA always makes the network more robust compared to an RA. We also study the resilience of RR with different parameters, as shown in Figure 3d. The system shows more resilience under the LA (dark points) in contrast to RA (light points). In addition, the average degree \(<k>\) and the number of active neighbours \(m\) both affect the width of the hysteresis region. The larger average degree, the wider the hysteresis region displayed by the system. These results will be helpful for studying the spontaneous recovery in real systems under LA. Next, the dynamic recovery network in an ER network is studied. Figure 4a shows \(z\) as a function of \(p^{*}\), and the results imply that these two simulated results under RA (light symbol) and LA (dark symbol) almost overlap. Similar to the RR network, there also exists discontinuous phase transitions, and the probability of external failure not only affects the value of the critical point but also affects the hysteresis region. We also compare the influence of the size of the ER network in Figure 4b. The fractions of active nodes as \(r=0.78\) for different \(N\) are almost same. As the parameters (\(r,p^{*}\)) change, the different initial node states can evolve into the hysteresis region, as shown in Figure 4c. Not surprisingly, the phase diagrams of the ER network under these two failure mechanisms almost show the same behaviours. This phenomena can also be found in a single static ER network [22]. Figure 4: Phase diagrams of the ER network. (**a**) Simulation results of the average fraction of nodes for three different \(r\), and the parameters are \(k=<10>,m=4\), and \(N=10^{5}\). (**b**) Simulation results of the average fraction of nodes for three different \(r\) values under LA for different \(N\) with \(<k>=10,m=4\), and \(r=0.78\). (**c**) The jump point for different parameters. Dark and light symbols denote LA and RA, respectively. Simulation results have been averaged over 1000 network realisations. Figure 3: Phase diagrams of the RR network. (**a**) Simulation results of the average fraction of nodes for three different \(r\) values under LA (dark symbols) and RA (light symbols). The parameters for the RR network are \(<k>=10,m=4\), and \(N=10^{5}\). (**b**) Simulation results of the average fraction of nodes for three different \(r\) values under LA for different \(N\) with \(<k>=10,m=4\), and \(r=0.78\). (**c**) The phase diagram in model parameters (\(r,p^{*}\)) exhibits two kinds of phase transition behaviours. Low- and high-activity correspond to the cases where the fraction of active nodes of the stationary state is small or large, respectively. Red points are the dividing points between the hysteresis region and high/low-activity regions under LA. Additionally, two bunches of red points merge at a critical point located at (\(r=0.63,p^{*}=0.45\)). Blue points are the similar results under RA. (**d**) Jump points for different parameters. Light points are RA, and dark points are LA. Simulation results have been averaged over 1000 network realisations. Scale-free networks also have similar phase transition behaviours, which are shown in Figure 5a-c. However, when the power index is \(\lambda=2.5\), the results both show second-order continuous phase transition behaviours under RA and LA. The proportion of internally failed nodes under different failure modes affects the proportion of average active nodes \(z\). As the internally failed nodes become smaller, the network system shows strong resilience under RA (light symbols) than LA (dark symbols). However, as \(p^{*}\) increases, the network system under LA shows more resilience than RA. With the increase of \(\lambda\), there is a first-order jump phenomenon for larger \(r\) values.