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Biomedical interaction networks have incredible potential to be useful in the prediction of biologically meaningful interactions, identification of network biomarkers of disease, and the discovery of putative drug targets. Recently, graph neural networks have been proposed to effectively learn representations for biomedical entities and achieved state-of-the-art results in biomedical interaction prediction. These methods only consider information from immediate neighbors but cannot learn a general mixing of features from neighbors at various distances. In this paper, we present a higher-order graph convolutional network (HOGCN) to aggregate information from the higher-order neighborhood for biomedical interaction prediction. Specifically, HOGCN collects feature representations of neighbors at various distances and learns their linear mixing to obtain informative representations of biomedical entities. Experiments on four interaction networks, including protein-protein, drug-drug, drug-target, and gene-disease interactions, show that HOGCN achieves more accurate and calibrated predictions. HOGCN performs well on noisy, sparse interaction networks when feature representations of neighbors at various distances are considered. Moreover, a set of novel interaction predictions are validated by literature-based case studies.