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This paper presents a systematic in-depth study on the existence, importance, and application of stable nodes in peer- to-peer live video streaming. Using traces from a real large-scale system as well as analytical models, we show that, while the number of stable nodes is small throughout a whole session, their longer lifespans make them constitute a significant portion in a per-snapshot view of a peer-to-peer overlay. As a result, they have substantially affected the performance of the overall system. Inspired by this, we propose a tiered overlay design, with stable nodes being organized into a tier-1 backbone for serving tier-2 nodes. It offers a highly cost-effective and deployable alternative to proxy-assisted designs. We develop a comprehensive set of algorithms for stable node identification and organization. Specifically, we present a novel structure, Labeled Tree, for the tier-1 overlay, which, leveraging stable peers, simultaneously achieves low overhead and high transmission reliability. Our tiered framework flexibly accommodates diverse existing overlay structures in the second tier. Our extensive simulation results demonstrated that the customized optimization using selected stable nodes boosts the streaming quality and also effectively reduces the control overhead. This is further validated through prototype experiments over the PlanetLab network.