Network models are designed to capture properties of empirical networks and thereby provide insight into the processes that underlie the formation of complex systems. As new information concerning network structure becomes available, it becomes possible to design models that more fully capture the properties of empirical networks. A recent advance in our understanding of network structure is the control profile, which summarizes the structural controllability of a network in terms of source nodes, external dilations, and internal dilations. Here, we consider the topological properties-and their formation mechanisms-that constrain the control profile. We consider five representative empirical categories of internal-dilation dominated networks, and show that the number of source and sink nodes, the form of the in- and out-degree distributions, and local complexity (e.g., cycles) shape the control profile. We evaluate network models that are sufficient to produce realistic control profiles, and conclude that holistic network models should similarly consider these properties.