Power-law networks such as the Internet, terrorist cells, species relationships, and cellular metabolic interactions are susceptible to node failures, yet maintaining network connectivity is essential for network functionality. Disconnection of the network leads to fragmentation and, in some cases, collapse of the underlying system. However, the influences of the topology of networks on their ability to withstand node failures are poorly understood. Based on a study of the response of 2,000 power-law networks to node failures, we find that networks with higher nodal degree and clustering coefficient, lower betweenness centrality, and lower variability in path length and clustering coefficient maintain their cohesion better during such events. We also find that network robustness, i.e., the ability to withstand node failures, can be accurately predicted a priori for power-law networks across many fields. These results provide a basis for designing new, more robust networks, improving the robustness of existing networks such as the Internet and cellular metabolic pathways, and efficiently degrading networks such as terrorist cells.