Anonymous messaging platforms, such as Secret, Yik Yak, and Whisper, have emerged as important social media tools for sharing one's thoughts without the fear of being judged by friends, family, or the public. Further, anonymous platforms like these are important in nations with authoritarian governments, where the right to free expression and sometimes the personal safety of the author of the message depend on anonymity. Whether for fear of judgment or personal endangerment, it is sometimes crucial to keep anonymous the identity of the user who initially posted a sensitive message. In this paper, we consider a global adversary who wishes to identify the author of a message; it observes either a snapshot of the spread of a message at a certain time or sampled timestamp metadata (or both). Recent advances in rumor source detection show that existing messaging protocols are vulnerable against such an adversary. We introduce a novel messaging protocol, which we call adaptive diffusion, and show that under the snapshot adversarial model, it spreads the messages fast and achieves a perfect obfuscation of the source when the underlying contact network is an infinite regular tree. That is, all users with the message are nearly equally likely to have been the origin of the message. Under the timestamp-based adversary, we show that it achieves optimal obfuscation asymptotically in the degree of the underlying regular tree. Experiments on a sampled Facebook network demonstrate that adaptive diffusion effectively hides the location of the source under both adversarial models, even when the graph is finite, irregular and has cycles.
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