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Network measures derived from empirical observations are often poor estimators of the true structure of system as it is impossible to observe all components and all interactions in many real world complex systems. Here, we study attack robustness of complex networks with data missing caused by: 1) a uniform random sampling and 2) a nonuniform random sampling. By introducing the subgraph robustness problem, we develop analytically a framework to investigate robustness properties of the two types of subgraphs under random attacks, localized attacks, and targeted attacks. Interestingly, we find that the benchmark models, such as Erdős-Rényi graphs, random regular networks, and scale-free networks possess distinct characteristic subgraph robustness features. We show that the network robustness depends on several factors including network topology, attack mode, sampling method and the amount of data missing, generalizing some well-known robustness principles of complex networks. Our results offer insight into the structural effect of missing data in networks and highlight the significance of understanding different sampling processes and their consequences on attack robustness, which may be instrumental in designing robust systems.

Subgraph Robustness of Complex Networks Under Attacks