Robustness of networks with dependency topology

The robustness of complex networks with dependency links has been studied in recent years. However, previous studies focused mostly on the robustness of networks with dependency relations having local and simple structures, not considering the general cases where global network topology is formed by dependency links. Here, we analyze the percolation properties of network models composed of both connectivity and dependency links, where in addition to the usual connectivity links, dependency links also follow a certain network topology. We perform theoretical analysis and numerical simulations to understand the critical effects of dependency topology on the network robustness. Our results suggest that for a given network topology of connectivity, dependency topology can influence the network robustness, leading to different percolation types. Furthermore, we also give the theoretical analysis and simulation results on different combinations of connectivity topology and dependency topology. Our results may help to design and optimize the network robustness considering the underlying complicated dependency relationships. Copyright c © EPLA, 2017 Introduction. – Components in critical infrastructures cannot function independently, and usually interact with others through connectivity or dependency links [1–15]. With the aid of connectivity links, nodes can function cooperatively as a network and will stop functioning either through their own failure or when they become disconnected from the giant component of the network. Dependency links show distinct functional relationships: if node A depends on node B, the failure of node B will cause node A to fail directly even if node A is still connected to the giant component [5,6]. For example, nodes in the power grid convey power flow to certain loads, whose faults may induce overloads and failures of other nodes. These successive failures can be represented as certain dependency link between nodes, i.e., failure spreads with a characteristic length [16]. Initiated by random failures or malicious attacks, these dependencies may lead to catastrophic events including (a)E-mail: daqingl@buaa.edu.cn blackouts in power grids and jamming in transportation networks [17,18]. Previous studies consider the network robustness with a simple dependency structure. Parshani et al. [5] introduced a network model having dependency groups with fixed size 2, see fig. 1(a). Bashan et al. [6] generalized Parshani’s results and also studied the effects of dependency groups whose sizes follow a normal distribution or a Poisson distribution. Meanwhile, dependency is also considered while studying the robustness of interdependent networks, as significant interactions exist between modern infrastructures [3,4,8,9,11,13,19–22]. Buldyrev et al. [3] developed a theoretical framework to understand the robustness of two coupled networks, and dependency is represented as a one-to-one correspondence between two networks, meaning that each node in one network depends on one and only one node in the other network and vice versa. Shao et al. [9] proposed a more general network model where interdependent networks may have multiple support-dependencies considering that a node in one