Microblogging websites, like Twitter and Weibo, are used by billions of people to create and spread information. This activity depends on various factors such as the friendship links between users, their topic interests and social influence between them. Making sense of these behaviors is very important for fully understanding and utilizing these platforms. Most prior work on modeling social-media either ignores the effect of social influence, or considers its effect only on link formation or post generation. In contrast, in this paper we propose POLIM, which jointly models the effect of influence on both link and post generation, leveraging weak supervision. We also give POLIM-FIT, an efficient parallel inference algorithm for POLIM which scales to large datasets. In our experiments on a large tweets corpus, we detect meaningful topical communities, celebrities, as well as the influence strengths patterns among them. Further, we find that there are significant portions of posts and links that are caused by influence, and this portion increases when the data focuses on a specific event. We also show that differentiating and identifying these influenced content benefits other quantitative downstream tasks as well, like predicting future tweets and link formation.