This paper proposes an analytical framework for peer-to-peer (P2P) networks and introduces schemes for building P2P networks to approach the minimum weighted average download time (WADT). In the considered P2P framework, the server, which has the information of all the download bandwidths and upload bandwidths of the peers, minimizes the weighted average download time by determining the optimal transmission rate from the server to the peers and from the peers to the other peers. This paper first defines the static P2P network, the hierarchical P2P network and the strictly hierarchical P2P network. Any static P2P network can be decomposed into an equivalent network of sub-peers that is strictly hierarchical. This paper shows that convex optimization can minimize the WADT for P2P networks by equivalently minimizing the WADT for strictly hierarchical networks of sub-peers. This paper then gives an upper bound for minimizing WADT by constructing a hierarchical P2P network, and lower bound by weakening the constraints of the convex problem. Both the upper bound and the lower bound are very tight. This paper also provides several suboptimal solutions for minimizing the WADT for strictly hierarchical networks, in which peer selection algorithms and chunk selection algorithm can be locally designed.
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