Named entity recognition is a fundamental task in natural language processing. Based on the sequence labeling paradigm for flat named entity recognition, multiple methods have been developed to handle the nested structures. However, they either require fixed recognition order or introduce complex hypergraphs. To tackle this problem, we propose a novel model named Local Hypergraph Builder Network (LHBN) that builds multiple simpler local hypergraphs to capture named entities instead of a single complex full-size hypergraph. The proposed model has three main properties: (1) The named entities that share boundaries are captured in the same local hypergraph. (2) The boundary information is enhanced by building local hypergraphs. (3) The hypergraphs can be built bidirectionally to take advantage of the identification direction preference of different named entities. Experiments illustrate that our model outperforms previous state-of-the-art methods on four widely used nested named entity recognition datasets: ACE04, ACE05, GENIA, and KBP17. The code is available at https://github.com/yanyk13/local-hypergraph-building-network.git.