A central goal in neuroscience is to understand how dynamic networks of neural activity produce effective representations of the world. Advances in the theory of graph measures raise the possibility of elucidating network topologies central to the construction of these representations. We leverage a result from the description of lollipop graphs to identify an iconic network topology in functional magnetic resonance imaging data and characterize changes to those networks during task performance and in populations diagnosed with psychiatric disorders. During task performance, we find that task-relevant subnetworks change topology, becoming more integrated by increasing connectivity throughout cortex. Analysis of resting state connectivity in clinical populations shows a similar pattern of subnetwork topology changes; resting scans becoming less default-like with more integrated sensory paths. The study of brain network topologies and their relationship to cognitive models of information processing raises new opportunities for understanding brain function and its disorders. Author Summary Our mental lives are made up of a series of predictions about the world calculated by our brains. The calculations that produce these predictions are a result of how areas in our brain interact. Measures based on graph representations can make it clear what information can be combined and therefore help us better understand the computations the brain is performing. We make use of cutting-edge techniques that overcome a number of previous limitations to identify specific shapes in the functional brain network. These shapes are similar to hierarchical processing streams that play a fundamental role in cognitive neuroscience. The importance of these structures and the technique is highlighted by how they change under different task constraints and in individuals diagnosed with psychiatric disorders.