Graphs have been widely used to represent objects and object connections in applications such as the web, social networks, and citation networks. Mining influence relationships from graphs has gained increasing interests in recent years because providing information on how graph objects influence each other can facilitate graph exploration, graph search, and connection recommendations. In this paper, we study the problem of detecting influence aspects, on which objects are connected, and influence degree (or influence strength), with which one graph node influences another graph node on a given aspect. Existing techniques focus on inferring either the overall influence degrees or the influence types from graphs. In this paper, we propose a systematic approach to extract influence aspects and learn aspect-level influence strength. In particular, we first present a novel instance-merging based method to extract influence aspects from the context of object connections. We then introduce two generative models, Observed Aspect Influence Model (OAIM) and Latent Aspect Influence Model (LAIM), to model the topological structure of graphs, the text content associated with graph objects, and the context in which the objects are connected. To learn OAIM and LAIM, we design both non-parallel and parallel Gibbs sampling algorithms. We conduct extensive experiments on synthetic and real data sets to show the effectiveness and efficiency of our methods. The experimental results show that our models can discover more effective results than existing approaches. Our learning algorithms also scale well on large data sets.