Internet-of-Thing systems have been treated as a novel platform for graph data acquisition. Contents like dynamic network topology, organization and control flows, and interactions among monitored objects all contribute to the huge volumes of graph data generated in IoTs. These data are believed to brought significant benefits to both the operation and functionalities of IoT systems, especially when combined with cutting-edge Artificial Intelligence techniques. However, these graph data are usually locally collected by data contributors with sensing devices, which could be both partially overlapped as they record same environment, and sensitive as they can indicate private physical status of contributors. Considering all challenges, current solutions for graph data collection in IoTs are incapable. Therefore, this paper proposes a novel framework for privacy-preserving distributed graph data collection for IoTs. The framework allows the graphs kept by data contributors to be partially overlapped, and can help the data broker to efficiently derive the universal view by combining these graphs. The differential privacy is applied for privacy preservation during data collection. The proposed problem aims at minimizing the total bandwidth consumption for graph collection, which is proved to be NP-complete. Then three algorithms are proposed for different circumstances, based on the diverse knowledge and purposes held by the data broker. Finally, both theoretical and numerical analysis have demonstrated the advancement of these methods.