Data extracted from social media platforms, such as Twitter, are both large in scale and complex in nature, since they contain both unstructured text, as well as structured data, such as time stamps and interactions between users. A key question for such platforms is to determine influential users, in the sense that they generate interactions between members of the platform. Common measures used both in the academic literature and by companies that provide analytics services are variants of the popular web-search PageRank algorithm applied to networks that capture connections between users. In this work, we develop a modeling framework using multivariate interacting counting processes to capture the detailed actions that users undertake on such platforms, namely posting original content, reposting and/or mentioning other users' postings. Based on the proposed model, we also derive a novel influence measure. We discuss estimation of the model parameters through maximum likelihood and establish their asymptotic properties. The proposed model and the accompanying influence measure are illustrated on a data set covering a five year period of the Twitter actions of the members of the US Senate, as well as mainstream news organizations and media personalities.