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Illuminating the interconnections between drugs and genes is an important topic in drug development and precision medicine. Currently, computational predictions of drug-gene interactions mainly focus on the binding interactions without considering other relation types like agonist, antagonist, etc. In addition, existing methods either heavily rely on high-quality domain features or are intrinsically transductive, which limits the capacity of models to generalize to drugs/genes that lack external information or are unseen during the training process. To address these problems, we propose a novel Communicative Subgraph representation learning for Multi-relational Inductive drug-Gene interactions prediction (CoSMIG), where the predictions of drug-gene relations are made through subgraph patterns, and thus are naturally inductive for unseen drugs/genes without retraining or utilizing external domain features. Moreover, the model strengthened the relations on the drug-gene graph through a communicative message passing mechanism. To evaluate our method, we compiled two new benchmark datasets from DrugBank and DGIdb. The comprehensive experiments on the two datasets showed that our method outperformed state-of-the-art baselines in the transductive scenarios and achieved superior performance in the inductive ones. Further experimental analysis including LINCS experimental validation and literature verification also demonstrated the value of our model.