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Heterogeneous information networks (HINs) are widely employed for describing real-world data with intricate entities and relationships. To automatically utilize their semantic information, graph neural architecture search has recently been developed on various tasks of HINs. Existing works, on the other hand, show weaknesses in instability and inflexibility. To address these issues, we propose a novel method called Partial Message Meta Multigraph search (PMMM) to automatically optimize the neural architecture design on HINs. Specifically, to learn how graph neural networks (GNNs) propagate messages along various types of edges, PMMM adopts an efficient differentiable framework to search for a meaningful meta multigraph, which can capture more flexible and complex semantic relations than a meta graph. The differentiable search typically suffers from performance instability, so we further propose a stable algorithm called partial message search to ensure that the searched meta multigraph consistently surpasses the manually designed meta-structures, i.e., meta-paths. Extensive experiments on six benchmark datasets over two representative tasks, including node classification and recommendation, demonstrate the effectiveness of the proposed method. Our approach outperforms the state-of-the-art heterogeneous GNNs, finds out meaningful meta multigraphs, and is significantly more stable.