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Easy to deploy, robust, and highly resilient to failures, epidemic algorithms are a potentially effective mechanism for propagating information in large peer-to-peer systems deployed on Internet or ad hoc networks. It is possible to adjust the parameters of epidemic algorithm to achieve high reliability despite process crashes and disconnections, packet losses, and a dynamic network topology. Although researchers have used epidemic algorithms in applications such as failure detection, data aggregation, resource discovery and monitoring, and database replication, their general applicability to practical, Internet-wide systems remains open to question. We describe four key problems: membership maintenance, network awareness, buffer management, and message filtering, and suggest some preliminary approaches to address them.