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We consider optimal attacks or immunization schemes on different models of random graphs. We derive bounds for the minimum number of nodes needed to be removed from a network such that all remaining components are fragments of negligible size.We obtain bounds for different regimes of random regular graphs, Erdős-Rényi random graphs, and scale free networks, some of which are tight. We show that the performance of attacks by degree is bounded away from optimality.Finally we present a polynomial time attack algorithm and prove its optimal performance in certain cases.
Balashov, N., Cohen, R., Haber, A., Krivelevich, M., & Haber, S. (2019). Optimal shattering of complex networks. Applied Network Science, 4(1). https://doi.org/10.1007/s41109-019-0205-5