Large language models (LLMs) are increasingly involved in shaping public understanding on contested issues. This has led to substantial discussion about the potential of LLMs to reinforce or correct misperceptions. While existing literature documents the impact of LLMs on individuals' beliefs, limited work explores how LLMs affect social networks. We address this gap with a pre-registered experiment (N = 1265) around the 2024 US presidential election, where we empirically explore the impact of personalized LLMs on belief accuracy in the context of social networks. The LLMs are constructed to be personalized, offering messages tailored to individuals' profiles, and to have guardrails for accurate information retrieval. We find that the presence of a personalized LLM leads individuals to update their beliefs towards the truth. More importantly, individuals with a personalized LLM in their social network not only choose to follow it, indicating they would like to obtain information from it in subsequent interactions, but also construct subsequent social networks to include other individuals with beliefs similar to the LLM -- in this case, more accurate beliefs. Therefore, our results show that LLMs have the capacity to influence individual beliefs and the social networks in which people exist, and highlight the potential of LLMs to act as corrective agents in online environments. Our findings can inform future strategies for responsible AI-mediated communication.