Optimization of network robustness to waves of targeted and random attacks

We study the robustness of complex networks to multiple waves of simultaneous (i) targeted attacks in which the highest degree nodes are removed and (ii) random attacks (or failures) in which fractions $p_t$ and $p_r$, respectively, of the nodes are removed until the network collapses. We find that the network design which optimizes network robustness has a bimodal degree distribution, with a fraction $r$ of the nodes having degree $k_2=(⟨k⟩-1+r)∕r$ and the remainder of the nodes having degree $k_1=1$, where $⟨k⟩$ is the average degree of all the nodes. We find that the optimal value of $r$ is of the order of $p_t∕p_r$ for $p_t∕p_r⪡1$.