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 <Subscript>1</Subscript> (close to the average number of links per node), and one node is of very large degree, <InlineEquation ID="Equ1"> <EquationSource Format="TEX">$k_2 \sim N^{2/3}$</EquationSource> </InlineEquation>, where N is the number of nodes in the network. Copyright Springer-Verlag Berlin/Heidelberg 2004
