About the Lifespan of Peer to Peer Networks

In this paper we analyze the ability of peer to peer networks to deliver a complete file among the peers. Early on we motivate a broad generalization of network behavior organizing it into one of two successive phases. According to this view the network has two main states: first centralized – few sources (roots) hold the complete file, and next distributed – peers hold some parts (chunks) of the file such that the entire network has the whole file, but no individual has it. In the distributed state we study two scenarios, first, when the peers are “patient”, i.e, do not leave the system until they obtain the complete file; second, peers are “impatient” and almost always leave the network before obtaining the complete file. We first analyze the transition from a centralized system to a distributed one. We describe the necessary and sufficient conditions that allow this vital transition. The second scenario occurs when the network is already in the distributed state. We provide an estimate for the survival time of the network in this state, i.e., the time in which the network is able to provide all the chunks composing the file. We first assume that peers are patient and we show that if the number of chunks is much less than e^n, where n is the number of peers in the network, then the expected survival time of the network is exponential in the number of peers. Moreover we show that if the number of chunks is greater than $rac{log n}{n+1}e^{n+1}$, the network's survival time is constant. This surprisingly suggests that peer to peer networks are able to sustain only a limited amount of information. We also analyze the scenario where peers are impatient and almost always leave the network before obtaining the complete file. We calculate the steady state of the network under this condition. Finally a simple model for evaluating peer to peer networks is presented.