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 〈 k 〉. 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 = [(〈 k 〉 - 1) / sqrt(〈) k 〉] sqrt(N) high degree nodes ("hubs") of degree sqrt(〈 k 〉 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.