Internet measurements have been a rich source for discovering power law relationships. However, many of these power law ``discoveries'' have turned out to be specious, especially with respect to connectivity-related measurements that are notorious for their ambiguities, incompleteness, and inaccuracies. This is bad news for many of the modeling efforts that have focused almost exclusively on reproducing the claimed power law relationships. In this talk, I will discuss an alternate modeling approach that is not tied to any claimed power law behavior and instead relies heavily on domain knowledge. This approach is capable of explaining a wide range of different system behaviors and provides a basis for exploring under which circumstances one should or should not expect power law relationships in Internet connectivity structures such as the Internet's router-level topology, AS graph, or overlay networks such as the Web or different Peer-to-Peer networks.