Optimization of Network Robustness to Random Breakdowns

We study network configurations that provide optimal robustness to random breakdowns for networks with a given number of nodes $N$ and a given cost--which we take as the average number of connections per node $kav$. We find that the network design that maximizes $f_c$, the fraction of nodes that are randomly removed before global connectivity is lost, consists of $q=[(kav-1)/sqrtkav]sqrt N$ high degree nodes (``hubs) of degree $sqrt{kav N}$ and $N-q$ nodes of degree 1. Also, we show that $1-f_c$ approaches 0 as $1/sqrt N$--faster than any other network configuration including scale-free networks. We offer a simple heuristic argument to explain our results.