Optimization of network robustness to random breakdowns

Abstract

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 ) / 〈 k 〉 ] N high degree nodes (“hubs”) of degree 〈 k 〉 N and N - q nodes of degree 1. Also, we show that 1 - f c approaches 0 as 1 / N —faster than any other network configuration including scale-free networks. We offer a simple heuristic argument to explain our results.