Social network modeling provides plenty of data but realistic models for network growth must be simple if any mathematical results are expected. We have used preferential attachment (PA) models with a small number of parameters in an attempt to strike a balance between the mathematics and the statistical fitting. The PA models struggle to match the data but provide a context in which to test methods and analyze estimation techniques. Numerical summaries of network characteristics are often estimated using methods imported from classical statistics without real justification. For example, the Hill estimator coupled with a minimum distance threshold selection technique are commonly used. We discuss some attempts to justify and understand these estimation methods in the context of PA models. Without a model and its properties, there is no way to understand the limitation of estimation methods.