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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.