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 hki. 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 1⁄4 1⁄2ðhki 1Þ= ffiffi h p ki ffiffiffiffi N p high degree nodes (‘‘hubs’’) of degree ffiffiffiffiffiffiffiffiffiffi hkiN p and N q nodes of degree 1. Also, we show that 1 f c approaches 0 as 1= ffiffiffiffi N p —faster than any other network configuration including scale-free networks. We offer a simple heuristic argument to explain our results. r 2006 Elsevier B.V. All rights reserved.