# Research on the Robustness of Interdependent Networks under Localized Attack
## 1 Introduction  
Nowadays, critical networked infrastructures, such as power grids, water supply networks, transportation networks, and the Internet, play an increasingly important role in modern society. These infrastructures are not isolated, but interact with each other through connections, such as material, energy, and information [1; 2; 3]. Those connections keep interdependent systems functional, but increase the vulnerability of systems under operational errors, aging, and even intentional disruption. Considering the significant loss caused by infrastructure failure, it is of great practical significance to study the robustness of interdependent networks, and this issue has become a hot topic in recent years. After [PERSON] et al. [4] proposed fully coupled interdependent networks in 2010, many scholars have conducted research on this issue. The common research paradigm is to construct interdependent networks under different coupling patterns based on different subnetworks, then analyze the influence of different features on the robustness of interdependent networks under different attack strategies. There are two kinds of subnetworks usually used, namely, artificial networks, such as Watts-Strogatz (WS), Erdos-Renyi (ER), Random Regular (RR), Barabasi-Albert (BA), and lattice networks [5; 6; 7; 8], and realistic infrastructure networks, such as power grids, water supply networks, the Internet [9], etc. According to the emphasis, the existing research can be divided into two aspects: one is to study the influence of the topology of subnetworks; the other is to analyze the influence of the dependencylinks between subnetworks on the robustness of interdependent networks. The topology of subnetworks, such as the degree of distribution [4, 8], clustering coefficient [10, 11, 12], and assortative [13], is an important factor affecting the robustness of interdependent networks, and most recent works are carried out from this perspective. Moreover, these studies are often carried out in the context of different attack strategies and coupling preference [8, 14, 15, 16, 17, 18]. Attack strategy is the method of selecting attack nodes, such as random attack (RA), targeted attack (TA) [16, 19], and localized attack (LA) [20, 21, 22, 23, 24]. Coupling preference refers to the pattern that the coupling links are established between subnetworks [25, 26], including assortative, disassortative, and random coupling, based on node degree or betweenness. The other factor affecting the robustness of interdependent networks is the dependency link, including coupling strength, link strength, and the direction of the link. First, interdependent networks can be divided into fully and partially-interdependent according to coupling strength, which means the partition of nodes with dependency links. This coupling strength leads to a change from a first-order to a second-order percolation transition at a critical threshold [27, 28, 29]. Next, strong and weak coupling can be identified according to the strength of the dependency link, or, for example, the probability of node failure after losing its dependency link, as there is a rich phase transition of interdependent networks when the strength of the dependency link changes [30]. Finally, the direction of the dependency link distinguishes directed and undirected interdependent networks [31, 32]. A directed interdependent system has poorer robustness performance than an undirected system, the main reason being that there are more possibilities for the existence of longer dependency chains in directed interdependent systems than undirected ones. Additionally, there are many other studies on this issue. Except the failure caused by a lack of a dependency link mentioned above, the failure caused by the excessive load of nodes or edges is also considered in the research of interdependent networks [17, 33]. Similar to topology, studies based on load failure are often carried out considering networks, coupling preference, and attack strategies.