While Large Language Models (LLMs) have been extensively tested in dyadic game-theoretic scenarios, their collective behavior within complex network games remains surprisingly unexplored. To bridge this gap, we present NetworkGames, a framework connecting Generative Agents and Geometric Deep Learning. By formalizing social simulation as a message-passing process governed by LLM policies, we investigate how node heterogeneity (MBTI personalities) and network topology co-determine collective welfare. We instantiate a population of LLM agents, each endowed with a distinct personality from the MBTI taxonomy, and situate them in various network structures (e.g., small-world and scale-free). Through extensive simulations of the Iterated Prisoner's Dilemma, we first establish a baseline dyadic interaction matrix, revealing nuanced cooperative preferences between all 16 personality pairs. We then demonstrate that macro-level cooperative outcomes are not predictable from dyadic interactions alone; they are co-determined by the network's connectivity and the spatial distribution of personalities. For instance, we find that small-world networks are detrimental to cooperation, while strategically placing pro-social personalities in hub positions within scale-free networks can significantly promote cooperative behavior. We validate the robustness of these findings through extensive stress tests across multiple LLM architectures, scaled network sizes, varying random seeds, and comprehensive ablation studies. Our findings offer significant implications for designing healthier online social environments and forecasting collective behavior. We open-source our framework to facilitate research into the social physics of AI societies.

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