A variety of tools are available for the analysis of social networks, and more are developed each day. However, published analyses of collaboration networks in particular tend either to rely on a basic suite of diagnostics or to be incorporated into proposals for new diagnostics. In our investigation into the research collaboration network of mathematics researchers, we began with the basic suite, including averages of degree, distance, and triadic closure, as well as degree–degree correlations (assortativity), across nodes. Deeper questions than these tools alone answer, such as whether common trends in similarly-sized subnetworks have common explanations and what accounts for any dissimilar behaviors, spurred us to make a more thorough and synthesized use of basic diagnostics than is typical of the literature. This led to several surprising results about, and a more robust understanding of, and the network topology and its evolution. Additionally, by carefully connecting the combinatorics behind a diagnostic and the real-world phenomena it measures, we have shown that subtleties in the combinatorial definition can result in profound differences in measurement. This paper surveys these results through several examples.