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Mesoscopic quantum systems exhibit complex many-body quantum phenomena, where interactions between spins and charges give rise to collective modes and topological states. Even simple, non-interacting theories display a rich landscape of energy states—distinct many-particle configurations connected by spin- and energy-dependent transition rates. The ways in which these energy states interact is difficult to characterize or predict, especially in regimes of frustration where many-body effects create a multiply degenerate landscape. Here, we use network science to characterize the complex interconnection patterns of these energy-state transitions. Using an experimentally verified computational model of electronic transport through quantum antidots, we construct networks where nodes represent accessible energy states and edges represent allowed transitions. We find that these networks exhibit Rentian scaling, which is characteristic of efficient transportation systems in computer circuitry, neural circuitry, and human mobility, and can be used to measure the interconnection complexity of a network. We find that the topological complexity of the state transition networks—as measured by Rent’s exponent— correlates with the amount of current flowing through the antidot system. Furthermore, networks corresponding to points of frustration (due, for example, to spin-blockade effects) exhibit an enhanced topological complexity relative to non-frustrated networks. Our results demonstrate that network characterizations of the abstract topological structure of energy landscapes capture salient properties of quantum transport. More broadly, our approach motivates future efforts to use network science to understand the dynamics and control of complex quantum systems.