Quantifying the Robustness of Network Controllability

In this paper, we propose closed-form analytic approximations for the minimum number of driver nodes needed to fully control networks, where links are removed according to both random and targeted attacks. Our approximations rely on the concept of critical links. A link is called critical if its removal increases the required number of driver nodes. We validate our approximation on both real-world and synthetic networks. For random attacks, the approximation is always very good, as long as the fraction of removed links is smaller than the fraction of critical links. For some cases, the approximation is still accurate for larger fractions of removed links. The approximation for an attack, where first the critical links are removed, is also accurate, as long as the fraction of removed links is sufficiently small. Finally, we show that the critical link attack is the most effective among 4 considered attacks, as long as the fraction of removed links is smaller than the fraction of critical links.