# Dynamical Recovery of Complex Networks under a Localised Attack
## 2 Model and Results
Figure 1: Schematic illustration of the dynamic recovery process under an LA. At time \(t-1\), after these four steps, the network changed into the fourth network in the first row. Nodes 1 and 3 are suffering internal failure, but node 3 has recovered one time (\(l=1\)), while node 1 just internally failed the last time (\(l=0\)). Additionally, node 2 is suffering an external failure (\(l=0\)). At time \(t\), nodes 4 and 5 are chosen to internally fail, and then nodes 7 and 8 only have one active neighbour, so they externally fail. Node 1, which internally fails at time \(t-1\), can recover one time (\(l=1\)), and node 3 has achieved \(l=2=\tau\), so it can be active again, and the edge between nodes 3 and 6 can also be active. Finally, node 2 reaches the time required for external recovery, and it becomes active (\(l=1=\tau^{*}\)). Next time, two nodes will be locally chosen again, and the node still belonging to the internal failure state can fail again, such as node 1 in the first network in the third row since its \(l\) changed to 0 again. Then, the process occurs repeatedly. Figure 2: Model dynamical recovery on the RR network for \(<k>=10,m=4,p^{*}=0.3,r=0.78\) with \(N=10^{5}\). (**a**) The high activity steady state (\(z(t)\) is about 0.685) for the increasing direction of \(p^{*}\) (from 0 to 1). (**b**) The low activity steady state (\(z(t)\) is about 0.17) for the decreasing direction of \(p^{*}\) (from 1 to 0). These results highlight that the fraction of active nodes at different failure stages displays a diver's steady state even though all the parameters are the same due to system's damaged memory. In Figure 3a, we show the fraction of the average active nodes \(z\) in a steady state as a function of \(p^{*}\) at three different \(r\) values for the RR network with \(<k>=10\) and \(m=4\) under a localised attack. As \(r=0.4\), the system has weak external failures, and the fraction of average active nodes \(z\) shows continuous phase behaviours for the increasing or decreasing directions of \(p^{*}\). As \(r\) equals 0.78 or 0.9 with strong external failures, the system \(z\) both shows dis-continuous phase behaviours for the increasing or decreasing directions of \(p^{*}\), in which the jump thresholds are different values. The hysteresis shown in Figure 3a is the characteristic feature of a first-order phase transition. Comparing the dynamics under RA (light symbols) and LA (dark symbols), it can easily be found that the critical point of LA is a bit larger than RA, which means that the RR network is more resilient under LA than RA. It is easy to explain because, for LA, only the internal failure nodes on the outer shell of the attacked hole will affect the external failure of nodes, but for RA, all internal failure nodes will affect the remaining active nodes. These results are also consistent with the results of a single static RR network [22]. In order to reduce the impact of the number of nodes, we also compare the average fraction of active nodes \(z\) when \(r=0.78\) under LA, which is shown in Figure 3b. The results imply that the impact of the network's finite size on the RR network is almost negligible, and the jump points are almost the same for different \(N=10^{3},10^{4},10^{5}\) when \(r=0.78\). By repeating this procedure for different \(r\) values, we obtain the two-parameter (\(r,p^{*}\)) phase diagram shown in Figure 3c. The jump points split the phase diagram into three regions: one is the high-activity region (purple), which means there are more active nodes, the other one is the low-activity region (pink), and the hysteresis region (green) is between the other two regions. One can observe that the initial condition (fraction of failed nodes) determines the final stable state, and if the fraction of failed nodes is smaller than one in the unstable state, the dynamic will reach the high-activity region. However, if there are more failed nodes, it will be easier to reach the low-activity region.