Graph structures are ubiquitous throughout the natural sciences. Here we develop an approach that exploits the quantum source's graph structure to improve learning via an arbitrary quantum neural network (QNN) ansatz. In particular, we devise and optimize a self-supervised objective to capture the information-theoretic closeness of the quantum states in the training of a QNN. Numerical simulations show that our approach improves the learning efficiency and the generalization behavior of the base QNN. On a practical note, scalable quantum implementations of the learning procedure described in this paper are likely feasible on the next generation of quantum computing devices.