  A central issue in complex networks is tolerance to random failures and
intentional attacks. Current literature emphasizes the dichotomy between
networks with a power-law node connectivity distribution, which are robust to
random failures but fragile to targeted attacks, versus networks with an
exponentially decaying connectivity distribution, which are less tolerant to
failures but more resilient to attacks. We prove analytically that the optimal
network configuration under a classic measure of robustness is altogether
different from both of the above: in all cases, failure and/or attack, there
are no more than three distinct node connectivities in the optimal network.
