Generating social networks is essential for many applications, such as epidemic modeling and social simulations. The emergence of generative AI, especially large language models (LLMs), offers new possibilities for social network generation: LLMs can generate networks without additional training or need to define network parameters, and users can flexibly define individuals in the network using natural language. However, this potential raises two critical questions: 1) are the social networks generated by LLMs realistic, and 2) what are risks of bias, given the importance of demographics in forming social ties? To answer these questions, we develop three prompting methods for network generation and compare the generated networks to a suite of real social networks. We find that more realistic networks are generated with "local" methods, where the LLM constructs relations for one persona at a time, compared to "global" methods that construct the entire network at once. We also find that the generated networks match real networks on many characteristics, including density, clustering, connectivity, and degree distribution. However, we find that LLMs emphasize political homophily over all other types of homophily and significantly overestimate political homophily compared to real social networks.