On the connectedness of peer-to-peer overlay networks

Peer-to-peer (P2P) applications typically use overlay networks to forward the content queries. However, due to the distributed fashion of the overlay establishment, the overlay networks in P2P applications are not guaranteed to be connected. In this paper, we introduce a new concept called ´buddy-assignment graph´ for the P2P overlay networks and prove a theorem that the connectedness of a P2P overlay network, being irrelevant to the kind of its overlay graph, is equivalent to the connectedness of its buddy-assignment graph. Then, with the aid of this theorem, we show that (1) when all hosts are active in the P2P overlay networks, two random buddies for each host suffice to make the connectedness probability close to 1 as well as approximating to 1 with the increase of the overlay scale and (2) when hosts are only active in certain probability, though the connectedness probability decreases with the increase of the overlay scale, the connectedness probability can still be very quickly raised to almost 1 by increasing the number of buddies for each host.