Here, we use simulation methods to examine how two features of the degree distribution affect network connectivity: Mean degree captures a volume dimension, while the skewness of the upper tail captures a shape dimension. We find a clear interaction between shape and volume: When mean degree is low, connectivity is greater for long-tailed distributions, but at higher mean degree, connectivity is greater in short-tailed distributions. The phase transition to a giant component and giant bicomponent emerges as a positive function of volume, but it rises more sharply and ultimately reaches more people in short-tail distributions than in long-tail distributions. These findings suggest that any interventions should be attuned to how practices affect both the volume and shape of the degree distribution, noting potential unanticipated effects. For example, policies that primarily affect high-volume nodes may not be effective if they simply redistribute volume among lower degree actors, which appears to exacerbate underlying network connectivity.