Efficient calculation of the robustness measure R for complex networks

In a recent work, Schneider et al. (2011) proposed a new measure R for network robustness, where the value of R is calculated within the entire process of malicious node attacks. In this paper, we present an approach to improve the calculation efficiency of R, in which a computationally efficient robustness measure R′ is introduced when the fraction of failed nodes reaches to a critical threshold qc. Simulation results on three different types of network models and three real networks show that these networks all exhibit a computationally efficient robustness measure R′. The relationships between R′ and the network size N and the network average degree 〈k〉 are also explored. It is found that the value of R′ decreases with N while increases with 〈k〉. Our results would be useful for improving the calculation efficiency of network robustness measure R for complex networks.