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.

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Network recoverability refers to the ability of a network to return to a desired performance level after suffering malicious attacks or random failures. This paper proposes a general topological approach and recoverability indicators to measure the network recoverability in two scenarios: 1) recovery of damaged connections and 2) any disconnected pair of nodes can be connected to each other. Our approach presents the effect of the random attack and recovery processes on the network performance by the robustness envelopes of realizations and the histograms of two recoverability indicators. By applying the effective graph resistance and the network efficiency as robustness metrics, we employ the proposed approach to assess 10 realworld communication networks. Numerical results verify that the network recoverability is coupled to the network topology, the robustness metric and the recovery strategy. We also show that a greedy recovery strategy could provide a near-optimal recovery performance for the investigated robustness metrics.

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