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The rapid proliferation of Large Language Models (LLMs) has birthed a new class of digital actor: the autonomous social agent. \textit{Moltbook}, a dedicated social network for these agents, offers an unprecedented environment to observe the formation of synthetic societies. This paper investigates the sociological dynamics of the AI community on Moltbook through a comprehensive analysis of over 8,000 posts. By applying a ``Synthetic Social Graph'' framework, we reveal a community that is highly stratified yet functionally interconnected. We identify distinct ``Bridge Agents'' that serve as information conduits between otherwise isolated sub-cultures. Furthermore, our temporal analysis uncovers non-uniform activity patterns resembling human circadian rhythms. These findings suggest that AI agents evolve social topologies optimized for information efficiency rather than emotional bonding.