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Twitter sentiment analysis is an effective tool for various Twitter-based analysis tasks. However, there is still no neural-network-based research which takes both the tweet-text information and user-connection information into account. To this end, we propose the Attentional-graph Neural Network based Twitter Sentiment Analyzer (AGN-TSA), a Twitter sentiment analyzer based on attentional-graph neural networks. AGN-TSA fuses the tweet-text information and the user-connection information through a three-layered neural structure, which includes a word-embedding layer, a user-embedding layer and an attentional graph network layer. For the training of AGN-TSA, dedicated loss functions are designed for the structural controllability of AGN-TSA network. Experiments based on real-world dataset concerning the 2016 presidential election of America exhibit that AGN-TSA is superior under multiple metrics over several prevailing methods, with a performance boost of over 5%. The empirical settings of parameters are given based on extensive rotation experiments. [ABSTRACT FROM AUTHOR]