Researchers have attempted to model information diffusion and topic trends and lifecycle on online social networks. They have investigated the role of content, social connections and communities, familiarity and behavioral similarity in this context. The current article presents a survey of representative models that perform topic analysis, capture information diffusion, and explore the properties of social connections in the context of online social networks. The article concludes with a set of outlines of open problems and possible directions of future research interest. This article is intended for researchers to identify the current literature, and explore possibilities to improve the art.