Dynamical robustness of networks based on betweenness against multi-node attack

We explore the robustness of a network against failures of vertices or edges where a fraction $f$ of vertices is removed and an overload model based on betweenness is constructed. It is assumed that the load and capacity of vertex $i$ are correlated with its betweenness centrality $B_i$ as $B_i^\theta$ and $(1+\alpha) B_i^\theta$ ($\theta$ is the strength parameter, $\alpha$ is the tolerance parameter). We model the cascading failures following a local load preferential sharing rule. It is found that there exists a minimal $\alpha_{\rm c}$ when $\theta$ is between 0 and 1, and its theoretical analysis is given. The minimal $\alpha_{\rm c}$ characterizes the strongest robustness of a network against cascading failures triggered by removing a random fraction $f$ of vertices. It is realized that the minimal $\alpha_{\rm c}$ increases with the increase of the removal fraction $f$ or the decrease of average degree. In addition, we compare the robustness of networks whose overload models are characterized by degree and betweenness, and find that the networks based on betweenness have stronger robustness against the random removal of a fraction $f$ of vertices.