A possible quadruple point in networks of directed networks under targeted attacks

Abstract

Many real systems are known to interact with one another, forming networks of networks (NONs). Plenty of attention has been poured into the research on the robustness in NONs in the past decade. Previous studies focus on undirected networks, or directed networks under random attacks. While many real networks are directed and how networks of directed networks (NODNs) respond to targeted attacks remains unknown. We thus develop a general analytical tool for analyzing the robustness of NODNs under two kinds of targeted attacks: degree-based attacks and in-degree (out-degree)-based attacks. The analytical tool can perfectly predict the sizes of the final giant strongly connected components and the phase transitions on the NODNs in response to targeted attacks. By applying the tool to synthesis networks, we find that a quadruple point intersected by four different phase regions could appear in the random regular NODNs. To the best of our knowledge, it is the first time that a quadruple point is found in the studies of complex networks. In addition, we find triple points intersected by three phases in networks of directed scale-free networks, and critical points that connect two phases in networks of directed Erdös–Rényi networks. The discovery of these tipping points could help understand network robustness and enable better design of networked systems.