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In this paper, we explore the relationship between the topological characteristics of a complex network and its robustness to sustained targeted attacks. Using synthesized scale-free networks, we look at a number of network measures, including rich club profiles, scale-free exponent, modularity, assortativity, average path length and clustering coefficient of a network, and how each of these influence the robustness of a scale-free network under targeted attacks. We consider sustained targeted attacks by order of node degree. We show that assortativity and average path length have a positive correlation with network robustness, whereas clustering coefficient has a negative correlation. We did not find any correlation between the modularity and robustness, scale-free exponent and robustness, or rich-club profiles and robustness. Our results highlight the importance of topological characteristics in influencing network robustness, and illustrate design strategies network designers can use to increase the robustness of scale-free networks under sustained targeted attacks.