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Self-supervised multiplex graph representation learning (SMGRL) has attracted increasing interest, but previous SMGRL methods still suffer from the following issues: (i) they focus on the common information only (but ignore the private information in graph structures) to lose some essential characteristics related to downstream tasks, and (ii) they ignore the redundant information in node representations of each graph. To solve these issues, this paper proposes a new SMGRL method by jointly mining the common information and the private information in the multiplex graph while minimizing the redundant information within node representations. Specifically, the proposed method investigates the decorrelation losses to extract the common information and minimize the redundant information, while investigating the reconstruction losses to maintain the private information. Comprehensive experimental results verify the superiority of the proposed method, on four public benchmark datasets.