Phase transitions in optimally robust network structures

If we add links to a network at random, a critical threshold can be crossed where a giant connected component forms. Conversely, if links or nodes are removed at random, the giant component shrinks and eventually breaks. In this paper, we explore which can optimally withstand random removal of a known proportion of nodes. When optimizing the size of the giant component after the attack, the network undergoes an infinite sequence of continuous phase transitions between different optimal structures as the removed proportion of nodes is increased. When optimizing the proportion of links in the giant component, a similar infinite sequence is observed, but the transitions are now discontinuous.