Social networks profoundly influence how humans form opinions, exchange information, and organize collectively. As large language models (LLMs) are increasingly embedded into social and professional environments, it is critical to understand whether their interactions approximate human-like network dynamics. We develop a framework to study the network formation behaviors of multiple LLM agents and benchmark them against human decisions. Across synthetic and real-world settings, including friendship, telecommunication, and employment networks, we find that LLMs consistently reproduce fundamental micro-level principles such as preferential attachment, triadic closure, and homophily, as well as macro-level properties including community structure and small-world effects. Importantly, the relative emphasis of these principles adapts to context: for example, LLMs favor homophily in friendship networks but heterophily in organizational settings, mirroring patterns of social mobility. A controlled human-subject survey confirms strong alignment between LLMs and human participants in link-formation decisions. These results establish that LLMs can serve as powerful tools for social simulation and synthetic data generation, while also raising critical questions about bias, fairness, and the design of AI systems that participate in human networks.