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A robust system should perform well under random failures or targeted attacks, and networks have been widely used to model the underlying structure of complex systems such as communication, infrastructure, and transportation networks. Hence, network robustness becomes critical to understanding system robustness. In this paper, we propose a spectral measure for network robustness: the second spectral moment m2 of the network. Our results show that a smaller second spectral moment m2 indicates a more robust network. We demonstrate both theoretically and with extensive empirical studies that the second spectral moment can help (1) capture various traditional measures of network robustness; (2) assess the robustness of networks; (3) design networks with controlled robustness; and (4) study how complex networked systems (e.g., power systems) behave under cascading failures. 

The widespread application of phasor measurement units has improved grid operational reliability. However, this has increased the risk of cyber threats such as false data injection attack that mislead time-critical measurements, which may lead to incorrect operator actions. While a single incorrect operator action might not result in a cascading failure, a series of actions impacting critical lines and transformers, combined with pre-existing faults or scheduled maintenance, might lead to widespread outages. To prevent cascading failures, controlled islanding strategies are traditionally implemented. However, islanding is effective only when the received data are trustworthy. This paper investigates two multi-objective controlled islanding strategies to accommodate data uncertainties under scenarios of lack of or partial knowledge of false data injection attacks. When attack information is not available, the optimization problem maximizes island observability using a minimum number of phasor measurement units for a more accurate state estimation. When partial attack information is available, vulnerable phasor measurement units are isolated to a smaller island to minimize the impacts of attacks. Additional objectives ensure steady-state and transient-state stability of the islands. Simulations are performed on 200-bus, 500-bus, and 2000-bus systems.