File-sharing networks are distributed systems used to disseminate files among a subset of the nodes of the Internet. A file is split into several pieces called chunks, the general simple principle is that once a node of the system has retrieved a chunk, it may become a server for this chunk. A stochastic model is considered for arrival times and durations of time to download chunks. One investigates the maximal arrival rate that such a network can accommodate, i.e., the conditions under which the Markov process describing this network is ergodic. Technical estimates related to the survival of interacting branching processes are key ingredients to establish the stability of these systems. Several cases are considered: networks with one and two chunks where a complete classification is obtained and several cases of a network with $n$ chunks.