Scale-free networks rely on a relatively small number of highly connected nodes to achieve a high degree of interconnectivity and robustness to random failure, but suffer from a high sensitivity to directed attack. In this paper we describe a parametrized family of networks and analyze their connectivity and sensitivity, identifying a network that has an interconnectedness closer to that of a scale-free network, a robustness to attack closer to that of an exponential network, and a resistance to failure better than that of either of those networks.