Robustness of scale-free networks with dynamical behavior against multi-node perturbation

An issue which is increasingly attracting attention from scientists to engineers, is the robustness of networks which is the ability against perturbations. It is found that both the network topology and network dynamics affect the robustness of networks. In this article, we present the cascading failure model triggered by perturbing a fraction $1-p$ of nodes on SF networks with three dynamics: the biochemical($B$), epidemic($E$) and regulatory($R$) dynamics. A mathematical method is developed to calculate the cascading failure size and the giant component to evaluate the robustness when a fraction $1-p$ of nodes is perturbed on SF dynamical networks. We perform extensive numerical simulations to test and verify this formula and find that the theoretical results are in good agreement with simulations. The results show that the network is more robust as the tolerance coefficient $\delta$ increases, and the size of network has little influence on the robustness, especially for $B$ and $R$. Remarkably, the heterogeneity of networks is positive on the robustness. Moreover, the different characteristics that as the parameter $B$ increases or the parameter $R$ decreases the network with $B$ is more robust, and as the parameter $R$ increases or the parameter $B$ decreases the network with $E$ and $R$ is more robust are found. These findings may be useful for engineers to improve the robustness of the network or design robust networks with dynamics.