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Efficient communication learning among agents has been shown crucial for cooperative multi-agent reinforcement learning (MARL), as it can promote the action coordination of agents and ultimately improve performance. Graph neural network (GNN) provide a general paradigm for communication learning, which consider agents and communication channels as nodes and edges in a graph, with the action selection corresponding to node labeling. Under such paradigm, an agent aggregates information from neighbor agents, which can reduce uncertainty in local decision-making and induce implicit action coordination. However, this communication paradigm is vulnerable to adversarial attacks and noise, and how to learn robust and efficient communication under perturbations has largely not been studied. To this end, this paper introduces a novel Multi-Agent communication mechanism via Graph Information bottleneck (MAGI), which can optimally balance the robustness and expressiveness of the message representation learned by agents. This communication mechanism is aim at learning the minimal sufficient message representation for an agent by maximizing the mutual information (MI) between the message representation and the selected action, and simultaneously constraining the MI between the message representation and the agent feature. Empirical results demonstrate that MAGI is more robust and efficient than state-of-the-art GNN-based MARL methods.