# On the Robustness of No-Feedback Interdependent Networks
## 2 Models  
### Interdependent Networks With No-Feedback Dependency  
There are two kinds of dependency relationships among interdependent networks: feedback and no-feedback [2; 3]. Figure 1a shows a model with undirected feedback dependency: \(A_{3}\) and \(B_{4}\) depend on each other, the same as \(B_{4}\) and \(A_{4}\). Figure 1b shows a model with directed feedback dependency: \(A_{4}\) depends on \(B_{4}\) and \(B_{4}\) depends on \(A_{3}\). In both Figure 1a,b, \(A_{3}\)-\(B_{4}\)-\(A_{4}\) constitutes a feedback relationship of dependency or an interdependent chain. Thus, the failure of node \(A_{3}\) can produce a feedback on other nodes in subnetwork \(A\) through node \(B_{4}\). Due to the existence of the interdependent chain, the robustness of feedback interdependent networks is usually very poor, and in extreme cases, even the failure of a single node will cause the complete collapse of the whole system [24]. In addition, nodes usually establish the dependency with nearest nodes rather than nodes far away from them. Suppose that nodes with the same number in different subnetworks are nearest from each other, the dependency links among nodes \(A_{3}\), \(B_{4}\) and \(A_{4}\) in Figure 1a,b are not consistent with the principle of nearest distance. Therefore, in order to improve the level of robustness, the feedback interdependent system shown in Figure 1a,b should be avoided. For interdependent networks composed of two subnetworks A and B, if node \(A_{i}\) depends on node \(B_{j}\) and \(B_{j}\) depends on \(A_{k}\), they must have \(k=i\), such a system is called the no-feedback interdependent system [43]. These kinds of interdependent networks were firstly constructed and the percolation process of such systems studied by [PERSON] in Reference [2]. In addition, there are few bidirectional (or undirected) dependency links similar to those shown as Figure 1a, and most of the dependency links in interdependent networks are directed [23]. For example, a node in the control network may control a power network node, but this power node is not necessarily responsible for the power supply of this control node, i.e., the power node depends on the control node but not vice versa, which constitutes unidirectional dependency. Generally speaking, a more general situation in the real world is that the dependency link has directionality and satisfies the constraint of no-feedback. An illustrative example is shown in Figure 1c. In this paper, the interdependent model with directed and no-feedback dependency was considered. Without the loss of generality, we assumed that a node from one subnetwork had no more than one dependent node from the other subnetwork. Many complex systems in the real world, e.g., communication and control networks that play the key role of connection in many interdependent systems, are usually modeled as random networks [7; 44]. Therefore, the Erdo-Renyi (ER) random network model is determined for the formulation of the  
Figure 1: Various interdependent networks models [43]. (**a**) Model with undirected and feedback dependency links; (**b**) Model with directed and feedback dependency links; (**c**) Model with directed and no-feedback dependency links. In sub-graph (**a**); node \(A_{3}\) and \(B_{4}\) depend on each other, same as node \(B_{4}\) and \(A_{4}\); in sub-graph (**b**); node \(A_{4}\) depend on \(B_{4}\), and \(B_{4}\) depend on \(A_{3}\). Both of them form a feedback dependency relationship. subnetwork. We assumed that both subnetworks had the same number of nodes (denote as \(N_{A}\), \(N_{B}\)) for simplicity. Sorting the node numbers of subnetworks A and B randomly, we then got two random sequences (\(A_{i1}\), \(A_{i2}\),..., \(A_{iN}\) and \(B_{j1}\), \(B_{j2}\),..., \(B_{jN}\)) and constructed the potential dependency relationship like \(A_{i1}\)-\(B_{j1}\), \(A_{i2}\)-\(B_{j2}\),..., \(A_{iN}\)-\(B_{jN}\) using one-to-one correspondence.