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The pervasive presence of social media greatly enriches online users' social activities, resulting in abundant social relations. Social relations provide an independent source for recommendation, bringing about new opportunities for recommender systems. Exploiting social relations to improve recommendation performance attracts a great amount of attention in recent years. Most existing social recommender systems treat social relations homogeneously and make use of direct connections (or strong dependency connections). However, connections in online social networks are intrinsically heterogeneous and are a composite of various relations. While connected users in online social networks form groups, and users in a group share similar interests, weak dependency connections are established among these users when they are not directly connected. In this paper, we investigate how to exploit the heterogeneity of social relations and weak dependency connections for recommendation. In particular, we employ social dimensions to simultaneously capture heterogeneity of social relations and weak dependency connections, and provide principled ways to model social dimensions, and propose a recommendation framework SoDimRec which incorporates heterogeneity of social relations and weak dependency connections based on social dimensions. Experimental results on real-world data sets demonstrate the effectiveness of the proposed framework. We conduct further experiments to understand the important role of social dimensions in the proposed framework.