  Networks with a given degree distribution may be very resilient to one type
of failure or attack but not to another. The goal of this work is to determine
network design guidelines which maximize the robustness of networks to both
random failure and intentional attack while keeping the cost of the network
(which we take to be the average number of links per node) constant. We find
optimal parameters for: (i) scale free networks having degree distributions
with a single power-law regime, (ii) networks having degree distributions with
two power-law regimes, and (iii) networks described by degree distributions
containing two peaks. Of these various kinds of distributions we find that the
optimal network design is one in which all but one of the nodes have the same
degree, $k_1$ (close to the average number of links per node), and one node is
of very large degree, $k_2 \sim N^{2/3}$, where $N$ is the number of nodes in
the network.
