# Dynamical Recovery of Complex Networks under a Localised Attack
## 1 Introduction  
Network science has attracted a lot of attention, and it can map the actual system into a network by treating individuals/entities as nodes and the relationship between individuals as edges [1; 2; 3]. There is much interest in the question of how to keep the resilience of a system after node or link failures. Previous studies mainly focused on the network structures [4; 5; 6; 7; 3], network dynamics [8; 9; 10; 11], and failure mechanisms [12; 13; 14; 15]. Most research on the network assumes that the network is static, while many real systems dynamically evolve. For example, traffic networks with flow changes over time [16], while friendship networks decreased in size with time increasing [17]. For some systems in real-world scenarios, after the collapse, they can spontaneously recover. For example, the transportation network returned to normal after a period of congestion. Additionally, the financial network may recover after most of its components fail from a financial crisis. To model these actual phenomena, [PERSON] et al. [12] developed a framework for understanding dynamic networks based on the node recovery process and found that they can lead to the spontaneous emergence of macroscopic phase-flipping phenomena. Based on this, the model is developed for interacting dynamical networks and shows a very rich phase diagram [14]. After that, the research on recovery networks became an important research topic and has been paid extensive close attention [18; 19; 20]. For example, [PERSON] proposed the recovery strategy for complex networks under random failures by considering that a fraction of the failed nodes are localised and recovered [21].  
However, most researches did not consider the dynamical recovery impact of different failure mechanisms and only studied the network under random failures. In real life, some natural disasters, such as earthquakes, floods, or military attacks on infrastructure networks, often spread from one node to the surroundings in turn [22]. This failure mechanism is named a localised attack (LA) [22; 23]. The main difference between an LA and random failure (RA) is that the former is only limited to a local area, while the latter is distributed throughout the whole network. This localised attack on some networks is more destructive than random failure [24; 25] because a cluster of nodes failing can trigger an avalanche. Previous studies mainly focused on static networks and did not consider dynamic networks under such failure. However, in real scenarios, such transient dynamics, such as failure propagation, and the recovery process take place by turn after suffering initial failure. Therefore, it is interesting and of practical application to study how to enhance the resilience of dynamic networks against localised attacks based on dynamical recovery.  
In this paper, the recovery model is based on three fundamental assumptions: (i) internal failure is independent of the neighbourhood, (ii) external failure is induced by failed neighbouring nodes if their number exceeds a threshold, and (iii) the implementation of spontaneous recovery after a period of time. Here, we mainly focus on the resilience behaviours through the above local failure propagation and recovery strategy.