Dynamical robustness of networks against multi-node attacked

A question in the robustness research of networks, which has not been addressed previously but may be more important and of wider interest, is how to consider spatio-temporal tolerance against failure propagation after a fraction f of nodes attacked. Here we develop a quantitative approach to examine the cascading overload condition based on the structure connectivity when a fraction f of nodes is attacked randomly. We also explore the critical threshold against cascading failures with two types of load redistribution rule. Fixing the value of β (the redistribution parameter) or τ (the initial load distribution parameter), we prove that the network shows the strongest robustness when the values of β is equal to τ, and the network robustness shows a growth trend with the decrease of f. We get a striking conclusion within the global load preferential sharing rule that the network robustness is independent of the network topology.