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Clustering-based methods, which alternate between the generation of pseudo labels and the optimization of the feature extraction network, play a dominant role in both unsupervised learning (USL) and unsupervised domain adaptive (UDA) person re-identification (Re-ID). To alleviate the adverse effect of noisy pseudo labels, the existing methods either abandon unreliable labels or refine the pseudo labels via mutual learning or label propagation. However, a great many erroneous labels are still accumulated because these methods mostly adopt traditional unsupervised clustering algorithms which rely on certain assumptions on data distribution and fail to capture the distribution of complex real-world data. In this paper, we propose the plug-and-play graph-based pseudo label correction network (GLC) to refine the pseudo labels in the manner of supervised clustering. GLC is trained to perceive the varying data distribution at each epoch of the self-training with the supervision of initial pseudo labels generated by any clustering method. It can learn to rectify the initial noisy labels by means of the relationship constraints between samples on the k Nearest Neighbor (kNN) graph and early-stop training strategy. Specifically, GLC learns to aggregate node features from neighbors and predict whether the nodes should be linked on the graph. Besides, GLC is optimized with 'early stop' before the noisy labels are severely memorized to prevent overfitting to noisy pseudo labels. Consequently, GLC improves the quality of pseudo labels though the supervision signals contain some noise, leading to better Re-ID performance. Extensive experiments in USL and UDA person Re-ID on Market-1501 and MSMT17 show that our method is widely compatible with various clustering-based methods and promotes the state-of-the-art performance consistently.