This paper models information diffusion in a network of Large Language Models (LLMs) that is designed to answer queries from distributed datasets, where the LLMs can hallucinate the answer. We introduce a two-time-scale dynamical model for the centrally administered network, where opinions evolve faster while the network's degree distribution changes more slowly. Using a mean-field approximation, we establish conditions for a locally asymptotically stable equilibrium where all LLMs remain truthful. We provide approximation guarantees for the mean-field approximation and a singularly perturbed approximation of the two-time-scale system. To mitigate hallucination and improve the influence of truthful nodes, we propose a reputation-based preferential attachment mechanism that reconfigures the network based on LLMs' evaluations of their neighbors. Numerical experiments on an open-source LLM (LLaMA-3.1-8B) validate the efficacy of our preferential attachment mechanism and demonstrate the optimization of a cost function for the two-time-scale system.