Recently, neural-symbolic architectures have achieved success on commonsense reasoning through effectively encoding relational structures retrieved from external knowledge graphs (KGs) and obtained state-of-the-art results in tasks such as (commonsense) question answering and natural language inference. However, these methods rely on quality and contextualized knowledge structures (i.e., fact triples) that are retrieved at the pre-processing stage but overlook challenges caused by incompleteness of a KG, limited expressiveness of its relations, and retrieved facts irrelevant to the reasoning context. In this paper, we present a novel neural-symbolic model, named Hybrid Graph Network (HGN), which jointly generates feature representations for new triples (as a complement to existing edges in the KG), determines the relevance of the triples to the reasoning context, and learns graph module parameters for encoding the relational information. Our model learns a compact graph structure (comprising both extracted and generated edges) through filtering edges that are unhelpful to the reasoning process. We show marked improvement on three commonsense reasoning benchmarks and demonstrate the superiority of the learned graph structures with user studies.