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We study the problem of representation learning for multiple types of entities in a co-ordered network where order relations exist among entities of the same type, and association relations exist across entities of different types. The key challenge in learning co-ordered network embedding is to preserve order relations among entities of the same type while leveraging on the general consistency in order relations between different entity types. In this paper, we propose an embedding model, CO2Vec, that addresses this challenge using mutually reinforced order dependencies. Specifically, CO2Vec explores indirect order dependencies as supplementary evidence to enhance order representation learning across different types of entities. We conduct extensive experiments on both synthetic and real world datasets to demonstrate the robustness and effectiveness of CO2Vec against several strong baselines in link prediction task. We also design a comprehensive evaluation framework to study the performance of CO2Vec under different settings. In particular, our results show the robustness of CO2Vec with the removal of order relations from the original networks. 

Learning the dependency relations among entities and the hierarchy formed by these relations by mapping entities into some order embedding space can effectively enable several important applications, including knowledge base completion and prerequisite relations prediction. Nevertheless, it is very challenging to learn a good order embedding due to the existence of partial ordering and missing relations in the observed data. Moreover, most application scenarios do not provide non-trivial negative dependency relation instances. We therefore propose a framework that performs dependency relation prediction by exploring both rich semantic and hierarchical structure information in the data. In particular, we propose several negative sampling strategies based on graph-specific centrality properties, which supplement the positive dependency relations with appropriate negative samples to effectively learn order embeddings. This research not only addresses the needs of automatically recovering missing dependency relations, but also unravels dependencies among entities using several real-world datasets, such as course dependency hierarchy involving course prerequisite relations, job hierarchy in organizations, and paper citation hierarchy. Extensive experiments are conducted on both synthetic and real-world datasets to demonstrate the prediction accuracy as well as to gain insights using the learned order embedding.