Optimization of robustness of scale-free network to random and targeted attacks

The scale-fee networks, having connectivity distribution $P(k)\sim k^{-\alpha}$ (where $k$ is the site connectivity), is very resilient to random failures but fragile to intentional attack. The purpose of this paper is to find the network design guideline which can make the robustness of the network to both random failures and intentional attack maximum while keeping the average connectivity $<k>$ per node constant. We find that when $<k>=3$ the robustness of the scale-free networks reach its maximum value if the minimal connectivity $m=1$, but when $<k>$ is larger than four, the networks will become more robust to random failures and targeted attacks as the minimal connectivity $m$ gets larger.