The study of interdependent networks has become a new research focus in recent years. We focus on one fundamental property of interdependent networks: vulnerability. Previous studies mainly focused on the impact of topological properties upon interdependent networks under random attacks, the effect of degree heterogeneity on structural vulnerability of interdependent networks under intentional attacks, however, is still unexplored. In order to deeply understand the role of degree distribution and in particular degree heterogeneity, we construct an interdependent system model which consists of two networks whose extent of degree heterogeneity can be controlled simultaneously by a tuning parameter. Meanwhile, a new quantity, which can better measure the performance of interdependent networks after attack, is proposed. Numerical simulation results demonstrate that degree heterogeneity can significantly increase the vulnerability of both single and interdependent networks. Moreover, it is found that interdependent links between two networks make the entire system much more fragile to attacks. Enhancing coupling strength between networks can greatly increase the fragility of both networks against targeted attacks, which is most evident under the case of max-max assortative coupling. Current results can help to deepen the understanding of structural complexity of complex real-world systems. Complex network is an important tool used to describe and analyze the structure and dynamical behaviors of complex systems [1] [2] [3] . Since real-world complex systems are becoming increasingly dependent on one another, the study of interdependent networks has become another new active topic in network science 4, 5 . Modern critical infrastructures are representative examples of interdependent systems, such as water supply, power stations, fuel supply, transportation, communication, etc. For example, considering the interdependence between power grids and communication networks, power grids need communication networks to transmit control signals and communication networks need power grids to provide power supply. The investigation of interdependent networks has led to new discoveries that cannot be explained using a single-network framework [6] [7] [8] [9] [10] [11] [12] [13] [14] . We focus on one fundamental property of interdependent networks: attack vulnerability. Albert et al. 15 raised the study of complex networks under attacks, they found the "robust yet fragile" generic property of scale-free networks: scale-free networks display an unexpected degree of robustness to random failures, however, these networks are extremely vulnerable to intentional attacks. Their research has triggered numerous theoretical and experimental works in this topic [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] . However, most previous studies mainly focused on single, isolated networks. Based on percolation theory, recently, Buldyrev et al. proposed a general framework to investigate the attack resilience of a system composed of two networks whose nodes are mutually dependent 26, 27 . In this model, attack on nodes is simulated by random node removal from one network. Due to the existence of interdependent links, an initial failure of only a small fraction of nodes in one network can lead to an iterative cascade of failures that cause both networks to become fragmented. Moreover, interdependent systems can react to random failures in a manner that is totally different from single networks, i.e., an interdependent system can exhibit a first-order (discontinuous) phase transition instead of the second-order (continuous) phase transition which is typical for single networks 16, 18 . 1
Gradually complex systems in the real world are becoming dependent on each other due to which interdependent networks has become an active part in network science REF .
Impact of Degree Heterogeneity on Attack Vulnerability of Interdependent Networks
Scientific Reports
Scientific Reports
Computer Science Medicine