*Result*: Phase shift optimization in reconfigurable intelligent surface-assisted UAV in hierarchical aerial computing networks.

*The evolution toward 6G wireless networks necessitates innovative solutions to support the massive Internet of Things (IoT) deployments with unprecedented computational and communication requirements, motivating this A comprehensive framework that integrates Reconfigurable Intelligent Surface (RIS) technology with hierarchical aerial computing networks by combining RIS-equipped Unmanned Aerial Vehicles (UAVs) operating as mobile edge computing nodes with High-Altitude Platforms (HAPs) to create a three-tier computing hierarchy addressing the limitations of conventional terrestrial infrastructure. The system model encompasses RIS-equipped UAVs serving terrestrial IoT devices with a single HAP providing high-capacity computational resources, where the RIS phase optimization is formulated as a Riemannian conjugate gradient problem on complex circle manifolds to maximize total system throughput while naturally handling unit modulus constraints through a three-stage sequential decomposition approach. Extensive Monte Carlo simulations demonstrate significant performance improvements over the comparable algorithm without RIS enhancement, with the RIS-enhanced system achieving 18% throughput improvement, near-linear scalability serving approximately 100% of available IoT devices compared to the algorithm In the comparable algorithm at 100 devices, a 95% task completion rate was maintained across all network loads versus 79–80% for the algorithm compared to the comparable algorithm. The results validate the potential of RIS-enabled aerial networks as a transformative solution for scalable and efficient 6G IoT services, with enhanced channel quality from intelligent phase configuration, enabling superior resource utilization and service provisioning in hierarchical computing architectures, establishing key contributions including novel RIS-aerial computing integration, advanced Riemannian manifold optimization with superior convergence properties, unified resource allocation combining stable matching theory with externality elimination, comprehensive performance analysis demonstrating practical viability, and real-world implementation considerations for future multi-UAV scenarios and energy-efficient designs. [ABSTRACT FROM AUTHOR]

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