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)/root < k >]root N high degree nodes (hubs) of degree root < k > N and N - q nodes of degree 1. Also, we show that 1-f(c) approaches 0 as 1/root N-faster than any other network configuration including scale-free networks. We offer a simple heuristic argument to explain our results. (c) 2006 Elsevier B.V. All rights reserved.