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We consider the impact of Byzantine attackers on peer-to-peer topologies for distributed storage using network coding. First, the problem is formulated as one of data flow in random evolving graphs, in which a data source and a data collector are connected to data keepers who may behave in a Byzantine fashion. We then derive analytical results for the probability of carrying out a successful distributed denial of service attack (that is, collecting contaminated information from the network), as well as the expected number of contaminated nodes at each timestep. Our results show that, even for a small number of Byzantine attackers in the network, the probability of collecting contaminated information is overwhelming, and that the dissemination of information by peers as opposed to a selected subset of nodes in the network increases the probability of contaminated information collection.