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The goal of zero-shot learning (ZSL) is to train a model to classify samples of classes that were not seen during training. To address this challenging task, most ZSL methods relate unseen test classes to seen(training) classes via a pre-defined set of attributes that can describe all classes in the same semantic space, so the knowledge learned on the training classes can be adapted to unseen classes. In this paper, we aim to optimize the attribute space for ZSL by training a propagation mechanism to refine the semantic attributes of each class based on its neighbors and related classes on a graph of classes. We show that the propagated attributes can produce classifiers for zero-shot classes with significantly improved performance in different ZSL settings. The graph of classes is usually free or very cheap to acquire such as WordNet or ImageNet classes. When the graph is not provided, given pre-defined semantic embeddings of the classes, we can learn a mechanism to generate the graph in an end-to-end manner along with the propagation mechanism. However, this graph-aided technique has not been well-explored in the literature. In this paper, we introduce the attribute propagation network (APNet), which is composed of 1) a graph propagation model generating attribute vector for each class and 2) a parameterized nearest neighbor (NN) classifier categorizing an image to the class with the nearest attribute vector to the image's embedding. For better generalization over unseen classes, different from previous methods, we adopt a meta-learning strategy to train the propagation mechanism and the similarity metric for the NN classifier on multiple sub-graphs, each associated with a classification task over a subset of training classes. In experiments with two zero-shot learning settings and five benchmark datasets, APNet achieves either compelling performance or new state-of-the-art results.