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News recommendation is one of the most challenging tasks in recommender systems, mainly due to the ephemeral relevance of news to users. As social media, and particularly microblogging applications like Twitter or Weibo, gains popularity as platforms for news dissemination, personalized news recommendation in this context becomes a significant challenge. We revisit news recommendation in the microblogging scenario, by taking into consideration social interactions and observations tracing how the information that is up for recommendation spreads in an underlying network. We propose a deep-learning based approach that is diffusion and influence-aware, called Influence-Graph News Recommender (IGNiteR). It is a content-based deep recommendation model that jointly exploits all the data facets that may impact adoption decisions, namely semantics, diffusion-related features pertaining to local and global influence among users, temporal attractiveness, and timeliness, as well as dynamic user preferences. To represent the news, a multi-level attention-based encoder is used to reveal the different interests of users. This news encoder relies on a CNN for the news content and on an attentive LSTM for the diffusion traces. For the latter, by exploiting previously observed news diffusions (cascades) in the microblogging medium, users are mapped to a latent space that captures potential influence on others or susceptibility of being influenced for news adoptions. Similarly, a time-sensitive user encoder enables us to capture the dynamic preferences of users with an attention-based bidirectional LSTM. We perform extensive experiments on two real-world datasets, showing that IGNiteR outperforms the state-of-the-art deep-learning based news recommendation methods.