This paper presents results from our measurement and modeling efforts on the large-scale peer-to-peer (p2p) overlay graphs spanned by the PPLive system which is arguably the most popular and largest multimedia streaming p2p system today. We believe that our findings can be used to understand large-scale p2p streaming systems for future planning of resource usage, and to provide useful and practical hints for future design of large-scale p2p streaming systems. Unlike other previous studies on PPLive, which focused on either network-centric or user-centric measurements of the system, our study is unique in (a) focusing on PPLive overlay-specific characteristics, and (b) being the first to derive mathematical models for its distributions of channel population size and session length. Our studies also reveal characteristics of multimedia streaming p2p overlays that are markedly different from existing file-sharing p2p overlays. Specifically, we find that: (1) Small PPLive overlays (as many as 500 nodes) are similar to random graphs in structure, (2) Average degree of a peer in the overlay (i.e., its out-degree) is independent of channel population size, (3) The availability correlation between PPLive peer pairs is bimodal, i.e., some pairs have highly correlated availability, while others have no correlation, (4) Unlike p2p file-sharing users, PPLive peers are impatient, (5) Session lengths (discretized, per channel) are typically geometrically distributed, (6) Channel Population Size variations are larger than in p2p file-sharing networks, yet they can be fitted with polynomial mathematical models. We conclude with a series of suggestions on how our findings can improve IPTV future design.