# On the Robustness of No-Feedback Interdependent Networks
## 1 Introduction  
The continuous well-adjusted operation of modern society is ubiquitous, however it should not be taken for granted. Many critical national infrastructure networks are indispensable, including energy networks, communication networks, and traffic networks. Most of these networks are not independent, i.e., the substance, energy, and information exchanges always exist, resulting in so-called interdependent networks. Evaluating the robustness of such systems is of great significance for the design of resilient infrastructures. However, analyzing the nature of interdependent networks with traditional complex network theory faces great challenges due to the dependencies that exist between these networks. In 2010, [PERSON] proposed a one-to-one correspondence and fully coupled interdependent networks model [1], and studied the cascading failure using percolation theory. Since then, investigation on the nature of interdependent networks has received wide popularity and a lot of dramatic advances have been made [2; 3; 4; 5; 6; 7; 8; 9; 10]. In the literature, the robustness of interdependent networks has been evaluated on the basis of three aspects: (1) the characteristics of subnetworks; (2) the property of dependency links; and (3) the failure mode of nodes. The characteristics of subnetworks include the number of subnetworks, degree distribution [11], assortative [12], and clustering coefficient [13; 14]. At present, many investigations have been conducted on double-layer interdependent networks, where the number of subnetworks is two. [PERSON] et al. [1] first evaluated the robustness of two layer and fully coupled interdependent networks and found that the percolation process of interdependent networks exhibited a first-order phase transition, which wasdifferent from the two-order phase transition of a single or isolated network. [PERSON] and [PERSON] et al. (2015, 2016) extended the double-layer model to a multi-layer and investigated the cascading failure process of interdependent networks consisting of three or more subnetworks. This type of network is known as the network of networks (NON). In addition, the influence of degree distribution on robustness has usually been studied based on different failure modes and coupling preference. For instance, it was concluded in Reference [17] that a broader degree distribution of subnetworks increases the robustness of random coupled interdependent networks that are under random attack. Another important factor that affects the robustness of interdependent networks is the property of dependency links, including coupling preference [18; 19; 20], directionality, and dependency strength. Coupling preference determines the way that dependency links are established between subnetworks, such as assortative coupling, disassortative coupling, and random coupling based on node degree or betweenness [21]. The dependency link can be either directed [22; 23] or undirected [13; 14]. [PERSON] et al. [24] investigated the influence of the dependency link direction on the robustness of interdependent networks, and found that the robustness of directed interdependent networks was worse than undirected ones. The dependency strength was defined as the failure probability of the node after losing its dependency link. [PERSON] et al. (2015) analytically demonstrated the influence of this factor on robustness with percolation theory and generating function. In terms of the failure modes of nodes, the majority of the previous work in this area used random or intentional attacks to represent natural failure or deliberate destruction based on node degree or betweenness [26]. Recently, new explanations on the meaning of failure modes have been proposed from the viewpoint of localized attack [27; 28] and fuzzy information attack [29].